Book of Abstract

Edited By Michel Bercovier Bertold Badler Professor of Scientific Computing

Leibniz Center For Research in Computer Science Edmund Landau Minerva Center for Research in Mathematical Analysis and Related Areas

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Scientific Committee Petter Bjørstad University of Bergen Roland Glowinski University of Houston Ronald Hoppe University of Augsburg Hideo Kawarada Ryutsu Keizai University David Keyes (Chair) Columbia University Ralf Kornhuber Free University of Berlin Yuri Kuznetsov University of Houston Ulrich Langer University of Linz Jacques Periaux CIMNE/UPC Barcelona Alfio Quarteroni Ecole Polytechnique Fédérale de Lausanne Zhong-Ci Shi Chinese Academy of Sciences Olof Widlund New York University Jinchao Xu Penn State University Organizing Committee Prof M.Bercovier (Chairman) The Hebrew University,School of Computer Science and Engineering Prof Amir Averbuch Institute of Mathematics and Computer Science, Tel Aviv University Prof Pinhas Z. Bar-Yoseph IACMM, Dept of Mechanical Engineering,Technion , Israel Prof Matania Ben-Artzi Institute of Mathematics, Hebrew University of Jerusalem Dr Micheal S.Engelman Corporate VP, Ansys Prof. Dan Givoli Dept. of Aerospace Engineering,Technion Prof Raz Kupferman Institute of Mathematics, Hebrew University of Jerusalem Prof Zohar Yosibash Dept of Mechanical Engineering, Ben Gurion University Managing Team: Prof M.Bercovier, Neva Teitsman (Administration) , Uri Heineman (Web, and Document Editing) Yehuda Arav , ( Tech Help) Diesenhaus Tours ,Orit, Anat, ( Registration, Reservations, Transfers )

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Contents

Invited Talks .................................................. 4

In Memoriam Moshe Israeli .......................... 13

Contributed Talks ........................................ 16

Minisymposuim Talks .................................. 36 1. Mini-Symposium 1.............................................. 37

2. Mini-Symposium 2a and 2b ............................... 42

3. Mini-Symposium 3.............................................. 47

4. Mini-Symposium 4.............................................. 51

5. Mini-Symposium 5.............................................. 56

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Part I

Invited Talks

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Principles of Systematic Upscaling (SU)

Achi Brandt

Weizmann Institute of Science and University of California-Los Angeles

Abstract:

Partly motivated by algebraic multigrid and renormalization group methods, systematic multiscale procedures are described for solving or simulating very large systems, with computations at each fine scale being confined to just small parts of the entire domain. Applicable to nonlinear, deterministic or stochastic, autonomous systems with strong interscale interactions. Examples from molecular dynamics to turbulent flows.

Analysis and Convergent Adaptive Solution of the Einstein Constraint Equations

Professor Michael Holst

Department of Mathematics University of California, San Diego

Abstract:

There is currently tremendous interest in geometric PDE, due in part to the geometric flow program used recently to solve the Poincare conjecture. Geometric PDE also play an expanding role in many other applications, such as understanding the gravitational wave models of Einstein. The need to validate these models has led to the construction of gravitational wave detectors in the last several years, such as the NSF-funded LIGO project. In this lecture, we consider the coupled nonlinear elliptic constraints in the Einstein equations, a geometric flow which describes the propagation of gravitational waves generated by collisions of massive objects such as black holes. The constraint equations must be solved numerically to produce initial data for gravitational wave simulations, and to enforce the constraints during dynamical simulations. In the first part of the lecture, we consider a thirty-year-old open question involving existence of solutions to the constraint equations on space-like hyper-surfaces with arbitrarily prescribed mean extrinsic curvature, and we give a partial answer using a priori estimates and a new type of topological fixed-point argument.

In the second part of this lecture, we develop some adaptive numerical methods for which we can prove a number of useful results on convergence, optimality, and scalability. Based on the a priori estimates developed in the first part of the talk, we first establish some critical discrete estimates. We then derive error estimates for Galerkin approximations, and describe a class of nonlinear approximation algorithms based on adaptive finite element methods (AFEM). We establish some new AFEM convergence and optimality results for geometric PDE problems with non-monotone nonlinearities such as the Einstein constraints. We then describe an overlapping domain decomposition algorithm based on the chart structure of the underlying

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domain manifold, and we outline a framework for establishing convergence of the algorithm.

We finish by illustrating the algorithms with some examples using the Finite Element ToolKit (FETK). Adaptive Multilevel Primal-Dual Interior-Point Methods in PDE

Constrained Optimization

Ronald H.W. Hoppe1,2 1 Dept. of Math., Univ. of Houston, Houston, TX 77204-3008, U.S.A.

2 Inst. of Math., Univ. of Augsburg, D-86159 Augsburg, Germany

Abstract:

We are concerned with structural optimization problems in CFD, life and material sciences where the state variables are supposed to satisfy a linear or nonlinear PDE and the design variables are subject to bilateral pointwise constraints. Within a primal-dual setting, we suggest an all-at-once approach based on interior-point methods. Coupling the inequality constraints by logarithmic barrier functions involving a barrier parameter and the PDE by Lagrange multipliers, the KKT conditions for the resulting saddle point problem represent a parameter dependent nonlinear system. The efficient numerical solution relies on multilevel path-following predictor-corrector techniques with an adaptive choice of the continuation parameter where the discretization is taken care of by finite elements with respect to nested hierarchies of simplicial triangulations of the computational domain. In particular, the predictor is a nested iteration type tangent continuation, whereas the corrector is a multilevel inexact Newton method featuring transforming null space iterations. We also discuss the application of model reduction techniques based on a decomposition of the computational domain.

As applications, we consider optimal shape designs of microfluidic biochips, electrorheological shock absorbers, and microstructured biomorphic ceramics.

Domain decomposition and electronic structure calculations: a new approach

Claude Le Bris

Ecole Nationale des Ponts et Chaussees and INRIA, France

Abstract:

Electronic structure calculations are a well known challenging problem of computational chemistry. Such calculations consist in nonlinear eigenvalue problems of large size, which have several specificities in comparison with other eigenvalue problems arising in other engineering sciences. The talk will first introduce the context, and point out the peculiarities of the problem under

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consideration. Then, a recently developed approach, based on the domain decomposition paradigm, will be presented.

The talk is based upon joint works with Maxime Barrault, Guy Bencteux (Electricite de France), Eric Cances (ENPC-INRIA) and William Hager (University of Florida). References include:

*Journal of Computational Physics, Volume 222, March 2007, pp~86-109, *Parallel Computing, to appear, *SIAM Numerical Analysis, submitted, http://hal.inria.fr/inria-00169080/

From Domain Decomposition to Homogeneisation in the Numerical Modelling of Materials

Patrick Le Tallec Ècole

Polytechnique, Dèpartement de Mècanique patrick.letallec@polytechnique.fr

Abstract:

Problems where there is a significant separation of scales between the global macroscopic problem and the local heterogeneities governing the response of the constitutive materials. They are based on the notion of representative volume elements (RVE), which are microscopic samples of the system under study. Each sample is solved at a microscopic scale taking as boundary conditions uniform or periodic displacement data deduced from the solution observed at macroscopic scale.

The talk will review some of these techniques and explain how to adapt mortar elements techniques as introduced in domain decomposition techniques to the construction of aposteriori error estimates in numerical homogeneization. The domain decomposition strategies turn out also to be useful for developing several approximate strategies for the numerical solution or the iterative coupling of the microscopic problems.

Variational scale separation methods

Jan Martin Nordbotten University of Bergen and Princeton

Abstract:

Many applications in the physical sciences require a detailed understanding of the interaction of processes on multiple scales. This has led to the development of countless analytical and numerical approaches, varying in generality and robustness. In this talk, we discuss and extend of some of the methods for dealing with multiple scales, seen from the context of potential driven conservation equations. These equations are relevant in modeling of

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e.g. heat transfer and stress-strain calculations for elastic materials; however, we will focus on the application to flow in porous media.

We will pay particular attention to the so-called multi-scale mixed finite elements, the mortar mixed finite elements, variational sub-grid upscaling, and their relationship to numerical linear algebra schemes such as balancing domain decomposition applied to the finest scale. This allows us to highlight desirable properties of the different methods, and ask the question: Can the best of these methods be combined?

We present an extension of the variational sub-grid upscaling method, which provides a general framework within which we discuss the remaining methods. While the development is given for the continuous problem, we nevertheless are able to construct a family of discrete multi-scale methods, within which existing methods appear as special cases.

We conclude the talk by comparing some of the discussed methods on example problems, in particular focusing on compressible two-phase flow in porous media. We address questions of accuracy, adaptivity, multiple scales, and computational cost.

Plane wave discontinuous Galerkin methods

Ilaria Perugia Dipartimento di Matematica Universita' di Pavia

Via Ferrata, 1 27100 Pavia – Italy ilaria.perugia@unipv.it

Abstract:

The oscillatory behavior of solutions to time harmonic wave problems, along with numerical dispersion, renders standard finite element methods inefficient already in medium-frequency regimes. As an alternative, several ways to incorporate information from the equation into the discretization spaces have been proposed in the literature, giving rise to methods based on shape functions which are solutions to either the primal or the dual problem, among them the so-called "ultra weak variational formulation" (UWVF) introduced by Despres for the Helmholtz equation in the early 1990's.

The UWVF is based on a domain decomposition approach and on the use of discontinuous piecewise plane wave basis functions. This method, which was numerically proved to be effective, has received a new interest very recently (see, e.g., papers by Huttunen, Kaipio, Malinen and Monk). From a theoretical point of view, the UWVF has been analyzed by Cessenat and Despres in 1998: they proved that the discrete solutions converge to the impedance trace of the analytical solution on the domain boundary. On the other hand, numerical results showed that convergence is achieved not only at the boundary, but in the whole domain.

In this talk, a priori convergence estimates in L2 and energy norms for the h-version of UWVF will be presented. The analysis is based on recasting the UWVF in the discontinuous Galerkin framework (a similar approach has been

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used very recently by Buffa and Monk for the derivation of L2 norm error estimates), and on new inverse and approximation estimates for plane waves in two dimensions, used in the context of duality techniques. Asymptotic optimality of the method in a mesh dependent energy norm and in the L2 norm are established. However, these estimates require a minimal resolution of the mesh to resolve the wavelength. Numerical evidence shows that this requirement which reflects the presence of numerical dispersion, cannot be dispensed with.

These results have been obtained in a joint work with Claude Gittelson and Ralf Hiptmair, Seminar for Applied Mathematics, ETH Zurich.

Numerical Zoom for Multi-Scale Problems

Olivier Pironneau LJLL-UPMC and Institut Universitaire de France

Olivier.Pironneau@upmc.fr

Abstract:

The safety assessment of a nuclear waste repository underground in a clay layer is by nature a multiscale problem for which, in principle it is not possible to obtain a numerical solution without extensive computer resources [3].

However if the solution is needed only in some small restricted region of space then a multiscale decomposition is possible which, when combine with a numerical zoom or domain decomposition, allows optimal precision with much less computer memory and cpu time.

The domain is decomposed into a large one where the simulations may not be precise and a small one where precision is required and the process can be iterated. A similar approach has been used before in Steger’s Chimera method [5]. In Brezzi et al [1] it was shown to be a particular implementation of Schwarz’ method and of Lions’ Hilbert space decomposition method [4]. Error estimates in the context of numerical zoom have been obtained by Wagner et al[6] and we will present here a better version for convergence of the iterative algorithm.

References: [1] Brezzi,F., Lions, J.L., Pironneau, O. (2001): Analysis of a Chimera Method. C.R.A.S., 332, 655-660 [2] J-B. Apoung-Kamga and J.L., Pironneau : O. Numerical zoom. DDM16 conference proceedings, New-York Jan 2005. David Keyes ed. [3] Del Pi˜no S. and O. Pironneau : Domain Decomposition for Couplex , The Couplex Exercise, Alain Bourgeat and Michel Kern. ed. (2003). [4] Lions, J.L., Pironneau, O. (1999): Domain decomposition methods for CAD. C.R.A.S., 328 73-80 [5] Steger J.L. (1991): The Chimera method of flow simulation. Workshop on applied CFD, Univ. of Tennessee Space Institute [6] Wagner J. : FEM with Patches and Appl. Thesis 3478, EPFL, March 2006.

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Domain Decomposition methods: industrial experience at

Hutchinson

François-Xavier Roux ONERA and University of Paris VI

Abstract:

In this presentation the unique experience of industrial utilisation of the FETI

methods at Hutchinson will be presented. In fact, after the theoretical basis of the method was established at the end of the eighties, an industrialisation, i.e. integration in a real finite element code was done in the framework of several European projects. In this very early stage of the development, the method was adopted by Hutchinson SA for its numerical simulation activities. Hutchinson is a world leader in manufacturing of rubber products for automotive, aerospace and other industries. Design of sophisticated components requires complex nonlinear simulations involving very large and ill conditioned models. In the middle of the nineties, the computer hardware available for Hutchinson was unable to tackle the models neither in term of speed, nor of storage. The only solution was to switch to a distributed computing environment, making the domain decomposition approach conjugated with clusters of PC as a good candidate for a breaking technology solution. Since then, the FETI method has proven itself as an extremely efficient simulation tool in real industrial environment. Currently, dozens of parallel jobs are run on a daily basis at Hutchinson Research Center and in several Hutchinson Departments. Static and dynamic rigidity, stress distribution in time or frequency domain, and other engineering quantities are computed for advanced metal-rubber components subject to large deformations and complex boundary conditions, such as contact with friction. Today, Hutchinson in-house finite element code contains several variants of one- and two-Lagrange Multiplier FETI solvers. During the presentation, several real industrial applications will be shown; problems of usage in industrial environment will be analyzed, and perspectives for both usage and code development will be outlined.

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The balancing domain decomposition methods by constraints (BDDC)

Professor Xuemin TU

University of California, Berkeley Abstract:

The balancing domain decomposition methods by constraints (BDDC) are non overlapping iterative substructuring domain decomposition methods for the solution of large sparse linear algebraic systems arising from the discretization of elliptic boundary value problems.

In this talk, the BDDC algorithms will be presented to solve a class of symmetric indefinite system of linear equations, which arises from the finite element discretization of the Helmholtz equation of time-harmonic wave propagation in a bound interior domain.

The proposed BDDC algorithm is closely related to the dual-primal finite element tearing and interconnecting algorithm for solving Helmholtz equations (FETI-DPH). The BDDC algorithms will also be presented to solve a class of non symmetric, positive definite linear systems resulting from the finite element discretization of advection-diffusion equations. In addition to the standard ubdomain vertex and edge/face average continuity constraints, certain flux average constraints across the subdomain interface, which depend on the coefficient of the first order term of the problems, will be introduced to improve the convergence.

The convergence rate estimate for the BDDC preconditioned GMRES iterations will be established and some numerical experiments will be discussed for both cases. This talk is based on a joint work with Professor Jing Li at Kent State University, U.S.A.

Accomodating Irregular Subdomains in Domain Decomposition Theory

Olof Widlund

Abstract:

In the theory for domain decomposition methods, we have previously often assumed that each subdomain is the union of a small set of coarse shape- egular triangles or tetrahedra. In this study, we discuss recent progress which makes it possible to analyze cases with irregular subdomains such as those provided by mesh partitioners

Our goal is to extend our analytic tools to problems on subdomains that might not even be Lipschitz and to characterize the rates of convergence of

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our methods in terms of a few, easy to understand, geometric parameters of the subregions. For two dimensions, we have already obtained some best possible results for scalar elliptic and linear elasticity problems: the subdomains should be John or Jones domains and the rate of convergence is determined using the parameters that define such domains and that of an isoparametric inequality. Progress on three dimensions will also be reported.

New results have also recently been obtained concerning variants of classical two-level additive Schwarz preconditioners. Our family of overlapping Schwarz methods borrows and extends coarse spaces from older iterative substructuring methods, i.e., methods based on non-overlapping subdomains. The local components of these preconditioners, on the other hand, are based on Dirichlet problems defined on a set of overlapping subdomains which cover the original domain.

Our methods are robust even in the presence of large changes, between subdomains, of the materials being modeled in the finite element models. An extra attraction is that our methods can be applied directly to problems where the stiffness matrix is available only in its fully assembled form.

We will also discuss several applications of the new tools. They include new results on almost incompressible elasticity and mixed finite elementsusing spaces of discontinuous pressures. We will also touch on recent work on Maxwell's equations in two dimensions.

Our work has been carried out in close collaboration with Clark R. Dohrmann of theSandia National Laboratories, Albuquerque, NM and Axel Klawonn and Oliver Rheinbach of the University of Duisburg-Essen, Germany.

Robust Iterative Methods for Singular and Nearly Singular

Systems of Equations

Professor Jinchao XU Penn. State University.

Abstract:

After giving a basic criterion for designing robust iterative methods (including multigrid and domain decomposition methods) based on space decomposition and subspace corrections, I will report a number of recent results on singular and nearly singular algebraic systems arising from the discretization of various (parameter- dependent) partial differential equations including elliptic equations with strongly discontinuous jumps, Maxwell equations, nearly incompressible linear elasticity, Stokes equations, Darcy's law and coupling of Stokes and Darcy.

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Part II

In Memoriam Moshe Israeli

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Contributions of Prof. Moshe Israeli to Scientific Computing

Amir Averbuch School of Computer Science

Tel Aviv University, Tel Aviv, Israel

Prof. Moshe Israeli and I have had strong collaborative relationships from 1986 until his sudden death on February 2007. This fruitful collaboration touches various fields related to scientific computing such as: parallel processing, asynchronous computation, domain decomposition methods for solving parabolic PDEs on multiprocessors, domain decomposition methods with local Fourier basis for parabolic problems, spectral multidomain techniques, parallel solutions of the Navier-Stokes equations, Poisson solvers (2D, 3D), time dependent diffusion equations with variable coefficients using multiwavelets, Helmholtz solvers, oscillatory integrals, material sciences, wavelet and multiresolution analysis, image processing, Integral transforms in pseudo- and polar grid including the Radon transform, modeling of nano-batteries.

I will give a short description of these areas and their contributions to scientific

computing and I will concentrate on one of our latest works: Irregular Sampling for Multidimensional Polar Processing of Integral

Transforms We describe a family of theories that enable to process polar data via integral

transforms. We show the relation between irregular sampling and discrete integral transforms, demonstrate the application of irregular (polar) sampling to image processing problems, and derive approximation algorithms that are based on unequally spaced samples. It is based on sampling the Fourier domain. We describe 2D and 3D irregular sampling geometries of the frequency domain, derive efficient numerical algorithms that implement them, prove their correctness, and provide theory and algorithms that invert them. We also show that these sampling geometries are closely related to discrete integral transforms. The proposed underlying methodology bridges via sampling between the continuous nature of the physical phenomena and the discrete nature world. Despite the fact that irregular sampling is situated in the core of many scientific applications, there are very few efficient numerical tools that allow robust processing of irregularly sampled data.

Joint works with R. Coifman, D. Donoho, Y. Shkolnisky.

Automated Transformations of PDE Systems Irad Yavneh

Department of Computer Science Technion

We study an approach for transforming systems of partial differential

equations (PDE) in order to obtain new formulations, especially decoupled ones that are more accessible to numerical solution. An algorithm is developed for generating such transformations automatically, using symbolic manipulations

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employing Groebner bases. The algorithm is implemented using freely available symbolic software. This approach, along with planned developments, will potentially provide a powerful set of tools for handling large systems of partial differential equations.

Joint work with Shmuel Onn Yossi Gil and Zvika Gutterman

Clustering Phenomena for Particulate Flow in Spinning Cylinders

Roland Glowinski, Department of Mathematics

University of Houston The main goal of this lecture is to present the result of numerical experiments addressing the direct numerical simulation of particle clustering phenomena taking place in a rotating cylinder containing a fluid-particle mixture, when the spinning angular velocity is large enough. After a brief description of the computational methodology used for these simulations, we will present movies showing how the spinning angular velocity influences the way particles clusters. A snapshot visualizing the transition to well separated clusters is shown on the figure below. This lecture is relevant to a mini-symposium dedicated to Moshe Israeli since: (i) The fluid component of the mixture is an incompressible viscous fluid whose flow is modeled by the Navier-Stokes equations. (ii) Laboratory experiments exhibiting this clustering phenomenon have been performed in the Chemical Engineering Department of the Israel Institute of Technology in Haiffa, the institution where our regretted colleague spent most of his very distinguished career.

Visualizations of the clustering phenomenon

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Part III

Contributed Talks

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Lagrange Sliding-Colliding Interfaces

Orna Agmon Ben-Yehuda Rafael Advanced Defence Systems LTD.

Abstract:

A Lagrange Sliding-Colliding Interface enables the free interaction of several meshes during the Lagrange Finite difference time step. It is based on dynamically identifying the boundary edges of the other meshes which may collide with a certain boundary vertex and identifying a "penetration point" for each such edge. From those non-additive possible collisions, the minimal sufficient set of interacting edges are chosen.

The interaction corrects the vertices' velocities to satisfy relative velocity constraints in order to prevent interpenetration while preserving the momentum. Later on, after the coordinates have been advanced to the next time step, the coordinates positions may be corrected (if the need arises).

The interface auto-detects the need for its existence - the user does not have to define it in advance.

Another usage of the interface is correcting small overlaps caused by independent re-mesh operations of several regions.

Distributed Hypersphere Decomposition in Arbitrary Dimensions

Aron J. Ahmadia*

Department of Applied Physics & Applied Mathematics, Columbia University

David E. Keyes Department of Applied Physics & Applied Mathematics, Columbia University

Abstract:

We describe a parallel technique for decomposing the surface of a hypersphere of arbitrary dimension, both exactly and approximately, into a specific number of regions of equal area and small diameter. We provide variations of the algorithm for when the number of regions does not need to be exact, and for when the absolute minimum diameter is not needed. We then analyze the performance of the various algorithms by finding the minimum distance between the centers of neighboring partitions, and use this to estimate the amount of work required to construct a partition with a minimum resolution for each algorithm. An accompanying C++ software package with a MATLAB interface will also be provided.

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Two-Level Schwarz Preconditioners for Super Penalty Discontinuous Galerkin Methods

Paola F. Antonietti*

Dipartimento di Matematica, Universit_a degli Studi di Pavia, via Ferrata 1, 27100, Pavia, Italy.

email: paola.antonietti@unipv.it

Blanca Ayuso Departamento de Matem_aticas, Universidad Aut_onoma de Madrid, Campus

de Cantoblanco, Ctra. de Colmenar Viejo, 28049, Madrid, Spain.

email: blanca.ayuso@uam.es

Abstract:

In recent years, much attention has been given to domain decomposition methods for linear elliptic problems that are based on a partitioning of the domain of the physical problem. Since the subdomains can be handled independently, such methods are very attractive for coarse-grain parallel computers. In this talk we shall present some non-overlapping additive and multiplica-tive Schwarz domain decomposition methods for the solution of the algebraic linear systems of equations arising from super penalty discontinuos Galerkin approximations of elliptic problems. We provide the convergence estimates, and we show that the proposed Schwarz methods can be successfully accelerated with suitable Krylov iterative solvers. Numerical experiments to validate our theory and to illustrate the performance and robustness of the proposed two-level methods will be presented.

Linearly implicit domain decomposition methods for

time-dependent reaction-diffusion equations

A. Arrarás†, L. Portero, J.C. Jorge Dpto. de Ingenieŕıa Matemática e Informática, Universidad Pública de

Navarra Campus de Arrosadía s/n, 31006, Pamplona (Navarra), Spain

{andres.arraras, laura.portero, jcjorge}@unavarra.es

Abstract:

This work is devoted to the numerical solution of a class of parabolic problems which involve nonlinear reaction and diffusion terms. The spatial discretisation is carried out using the support-operator method (cf. [3]), which constructs a cellcentred finite difference scheme that is naturally adapted to non-Cartesian grids. The resulting nonlinear system of stiff differential equations is then integrated in time by means of a linearly implicit fractional step method which extends the ideas proposed in [2] to the nonlinear case. At each time step, the discrete nonlinear diffusion operator is expressed as the sum of two components: one of them groups the first two terms of its Taylor

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expansion around the numerical solution at the previous time step, whereas the other involves the remaining higher-order terms of that expansion. The former is a discrete linear operator, which is subsequently split into a number of simpler linear suboperators, whilst the latter remains nonlinear. Such operator splitting is subordinated to a decomposition of the spatial domain into a set of overlapping subdomains in order to obtain an efficient parallel algorithm (cf. [1]). Finally, the linearly implicit method considers implicit time integrations for the linear suboperators while explicitly handling both the remaining discrete nonlinear operator and the discrete nonlinear reaction term. Therefore, the original problem is reduced to the solution of several linear systems per time step which can be trivially decomposed into a set of uncoupled parallelisable linear subsystems. As a difference with classical domain decomposition techniques, our proposal does not require any Schwarz iterative procedure. Numerical results illustrate a second-order convergence in both space and time for the described algorithm. References: [1] L. Portero, B. Bujanda, J.C. Jorge. A combined fractional step domain decomposition method for the numerical integration of parabolic problems. Lecture Notes in Comput. Sci. 3019 (2004), 1034–1041. [2] L. Portero, J.C. Jorge. A generalization of Peaceman–Rachford fractional step method. J. Comput. Appl. Math. 189 (2006), 676–688. [3] M. Shashkov. Conservative finite-difference methods on general grids. CRC Press, Boca Raton, 1996. †Corresponding author.

NKS for Fully Coupled Fluid-Structure Interaction Problems

Andrew Barker Department of Applied Mathematics

University of Colorado at Boulder Boulder, CO 80309

andrew.barker@colorado.edu

Xiao-Chuan Cai* Department of Computer Science University of Colorado at Boulder

Boulder, CO 80309 cai@cs.colorado.edu

Abstract:

We study a parallel Newton-Krylov-Schwarz method for solving

systems of nonlinear equations arising from the fully coupled, implicit finite element discretization of fluid-structure interaction problems on unstructured dynamic meshes. As expected the coupled system is considerably harder to solve than the individual fluid system or the solid system, but we show that the Schwarz preconditioner is capable of reducing the coupling effect and therefore guarantees the fast and scalable convergence of the Krylov subspace method.

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We focus on the simulations of blood flows in compliant arteries in 2D modeled by the coupled elastic wave equation and the incompressible Navier-Stokes equations. Numerical results obtained on parallel computers with hundreds of processors will be reported.

UNIFORM PRECONDITIONING FOR GENERALIZED FINITE ELEMENT METHOD DISCREITATIONS AND ITS APPLICATION

JAMES BRANNICK

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA (brannick@math.psu.edu)

DURKBIN CHO*

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA (cho@math.psu.edu)

JINCHAO XU Department of Mathematics, The Pennsylvania State University, University

Park, PA 16802, USA; And Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing 100871, China (xu@math.psu.edu)

LUDMIL ZIKATANOV

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA (ludmil@psu.edu).

Abstract:

This poster is on the e_cient solution of linear systems arising in discretizations of second order elliptic PDEs by a generalized _nite element method (GFEM). The multigrid methods with line Gauss-Seidel smoothers apply for GFEM equations on uniform rectangular grids in 2 spatial dimension. We prove that the resulting multigrid methods converge uniformly. Our further results apply for GFEM equations on unstructured simplicial grids in 2 and 3 spatial dimensions. We propose an e_cient preconditioner by using auxiliary space techniques and an additive preconditioner for the auxiliary space problems. We also prove that the condition number of the preconditioned system is uniformly bounded with respect to the mesh parameters. These results have a potential application in the design of a multilevel preconditioner for the pure traction problem of linear elasticity.

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Efficient simulation of multi-body contact problems on complex geometries

T. Dickopf, R. Krause

Institute for Numerical Simulation, University of Bonn

Abstract:

In this talk we consider the numerical simulation of non-linear contact problems in elasticity on complex three-dimensional geometries. In the case of curvilinear contact boundaries and non-matching finite element meshes, particular emphasis has to be put on the discretization of the transmission of forces and the non-penetration conditions at the contact interface.

Expressing the discrete contact constraints in a weak sense by means of a non-conforming domain decomposition method allows for a proof of the optimality of the discretization error. However, the computation of the discrete transfer operator requires great effort in implementation and additional analysis.

We develop an efficient method to assemble the discrete coupling operator, which can be regarded as a global approximation of a contact mapping by a composition of local projections and inverse projections. The emerging non-linear system can be solved efficiently by a monotone multigrid method.

We illustrate the effectiveness of our approach by several numerical examples in 3D and consider a biomechanical application. More precisely we present a possible strategy for the examination of the biomechanics of the spine, which has been altered due to the implantation of an artificial disc.

TaGas-Granular Flow Simulation by Parallel Computation

Joseph Falcovitz* Institute of Mathematics,

The Hebrew University of Jerusalem, Israel

Eran Kot Department of Physics, Tel Aviv University, Israel

David Sidilkover

Propulsion Physics Division, Soreq NRC, Israel

Abstract:

We present a recent stage in an ongoing research effort aimed at achieve a realistic simulation of gas--granular {\em heterogeneous} flow, such as the process that takes place upon igniting a granular charge in a gun chamber. This methodology, named Lagrange-Euler-Gas-Granular-Simulation (LEGGS), is based on explicit gas-to-grain coupling (momentum and heat exchange), as well as grain-to-grain coupling (collisions). A physically sound and accurate description of gas-grain flows with combustion (or even

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particles-gas 2-phase combustion) can be obtained by LEGGS methodology in complex geometries. The subject of the current presentation is a parallel 3D gas dynamics code based on the GRP scheme.

At the beginning of this work we were faced with the usual dilemma -- how to combine an accurate representation of moving waves (including shocks) with the capability of treating complex geometries--. The configuration we adopted relies on a Cartesian grid, using cut-cells to accommodate contoured boundary surfaces. Test cases involving a shock diffraction past a sphere and a Laval nozzle flow are presented. The computational results are validated by comparison to experimental data or to analytical solutions. Plans for future development will be discussed.

Computational Tool for a Mini-Windmill study with SOFT

M. Garbey*, M. Smaoui, N. De Brye, and C. Picard Abstract:

In this paper we present a new concept leading to the completely automated conception of a Vertical Axis Wind Turbine (VAWT). Our tool, SOFT, goes through four steps: Simulation, Optimum design, Fabrication and Testing. It works as follows:

1. We do a computational fluid dynamic simulation of the fluid structure interaction flow problem. We derive from that computation the torque and average rotation speed for a given friction coefficient on the rotor shaft and average flow speed. Our objective function is to get the most power out of the windmill.

2. We optimize the shape of the blade wingspan based on a surface response or eventually a genetic algorithm. The objective function corresponds to a direct simulation of the Navier Stokes flow interacting with the rotating turbine, until a steady regime is reached. This simulation takes time and we distribute the computation of the evaluation function for each gene on a network of computers.

So far this procedure is fairly standard in optimum design of turbine or wing shape. The next two steps are more innovative:

3. From the result of the optimization procedure we get a supposedly optimum shape. This shape is sent to a three D printer that fabricates the turbine. This turbine is mounted on a standardize base that has an electric generator.

4. The windmill is tested in a mini wind tunnel. We measure the rotation speed and power output with an electronic tester. This information is analyzed by the computer system and compared to the simulation.

5. The system can then decide to refine the simulation or to restart the SOFT loop for a different class of design, depending on the number of blades, the number of stages in the turbine, the use of a stator to channel the flow etc…

This project has some obvious pedagogic components that can motivate undergraduate students to do science!

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However, in reality, a critical step in the process is obviously the CFD method to test the VAWT performance: the numerical simulator should be robust, extremely fast but accurate enough to discriminate bad design from good design.

We will discuss an immersed boundary method and domain decomposition solver that we have tentatively developed to satisfy this ambitious program.

Component-Averaged Domain Decomposition Techniques

Dan Gordon*

Dept. of Computer Science, University of Haifa, Haifa, Israel.

Email: gordon@cs.haifa.ac.il

Rachel Gordon Dept. of Aerospace Engineering, the Technion - Israel Inst. of Technology,

Haifa, Israel Email: rgordon@tx.technion.ac.il

Abstract:

Component-Averaged Domain Decomposition (CADD) was introduced by Gordon and Gordon through the CARP algorithm and its CG acceleration CARP-CG. In CADD, external grid points of a subdomain are "cloned" (copied) into the subdomain, and the clones are updated by the subdomain solver together with the subdomain's internal points. The final values of all boundary points are taken as the average of their updated values and their clones in neighboring subdomains; this differs from standard DD methods. In CARP and CARP-CG, Kaczmarz row projections are performed in each subdomain, and then the results from the different subdomains are merged by the CADD method. CARP-CG is extremely robust and efficient on stiff elliptic PDE problems. It also produced excellent results in electron tomography, and it has shown a good potential for CFD problems with unstructured grids.

The robustness of CARP-CG on grids of varying sizes indicates its potential usefulness for multilevel applications.

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Numerical study on a BDDC algorithm for mortar discretizations of elasticity problems

Hyea Hyun Kim

Department of Mathematics, Chonnam National University, Gwangju 500-757, Korea

Hope page: http://www.math.jnu.ac.kr/hhkim/ Abstract:

A BDDC algorithm with the Neumann-Dirichlet preconditioner has been developed for elliptic problems and shown to be the most efficient for the problems with high contrast ratio of coefficients across sub interfaces.

The algorithm was extended to three dimensional elasticity problems with heterogeneous material parameters on geometrically non-conforming subdomain partitions.

Numerical experiments on the BDDC algorithm for the elasticity problems will be presented. The performance of the method regarding to discontinuous material parameters, to bad aspect ratio of interfaces, and to the selection of primal and non-primal interfaces will be investigated.

A preconditioner for generalized saddle point problems with

an indefinite block.

Piotr Krzyzanowski University of Warsaw, ul. Banacha 2, 02-097 Warszawa,

Poland. Email: piotr.krzyzanowski@mimuw.edu.pl Abstract:

Recently there have been developed several preconditioning methods for systems of block structure: A & B^T B & 0

where A is symmetric and positive definite. Some of these methods used block preconditioners in order to apply the existing domain decomposition preconditioners in this context. We discuss the construction and properties of preconditioners for symmetric systems as above, but in the case when the A block is indefinite and possibly singular. An example of such a problem is the discrete time-harmonic Maxwell's equation with large wave number. We will show the conditions under which the block diagonal and block triangular preconditioners bulit up from any good preconditioner for the corresponding s.p.d. subproblems are optimal with respect to the mesh size. We shall also discuss similarities of this approach to the augmented Lagrangian technique.

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Lower bounds for eigenvalues of elliptic operators by overlapping domain decomposition

Yuri Kuznetsov

Professor, Department of Mathematics, University of Houston, USA Abstract:

In this presentation, we consider a new approach to the estimation from below the lowest eigenvalues of symmetric positive definite elliptic operators. The approach is based on the overlapping domain decomposition procedure and on the replacement of subdomain operators by the special low rank perturbed scalar operators. The algorithm is illustrated by applications to the elliptic problems with mixed boundary conditions and strongly discontinuous coefficients.

Parallel Interface Concentrated Finite Element

Tearing and Interconnecting Methods

Ulrich Langer*1,2,3, Sven Beuchler1,2, Clemens Pechstein2 1 Institute of Computational Mathematics,

Johannes Kepler University Linz, 2 SFB F013, Johannes Kepler University Linz

3 RICAM, Austrian Academy of Sciences Altenbergerstr. 69, A-4040 Linz, Austria

e-mail: ulanger@numa.uni-linz.ac.at Abstract:

This talk is devoted to the fast solution of Interface-Concentrated (IC) finite element equations. The IC finite element schemes are constructed on the basis of a non-overlapping domain decomposition where a conforming boundary concentrated finite element approximation is used in every subdomain. Similar to boundary element domain decomposition methods the total number of unknowns per subdomain behaves like O((H/h)d−1), where H, h, and d denote the usual scaling parameter of the subdomains, the average discretization parameter of the subdomain boundaries, and the spatial dimension, respectively. We propose and analyze tearing and interconnecting methods which asymptotically exhibit the same complexity as the number of unknowns up to a logarithmic factor. In particular, our IC Finite Element Tearing and Interconnecting (FETI) solver is highly parallel and robust with respect to large coefficient jumps. We present numerical results confirming the parallel efficiency and the robustness of our IC-FETI solver, and we compare the IC-FETI with the standard one-level FETI solvers.

The authors gratefully acknowledge the financial support of the FWF (Austrian Science Funds) Special Research Program SFB F013 on “Numerical and Sympolic Scientific Computing”.

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A Neumann-Neumann algorithm for mortar finite element discretization of fourth order problems

Leszek Marcinkowski

Department of Mathematics University of Warsaw

Banacha 2, 02-097 Warszawa, Poland lmarcin@mimuw.edu.pl

http://www.mimuw.edu.pl/~lmarcin/

Abstract:

The purpose of the talk is to present a Neumann-Neumann type of method for solving a system of linear equations arising from the discretization of a model fourth order boundary value problem with discontinuous coefficients on locally nonconforming meshes.

The coefficients can be discontinuous with possibly very large jumps across the boundaries of the subregions of the original domain.

The finite element discretization is done by triangulating independently each subdomain and then the final discretization is built by a mortar technique, i.e. local conforming finite element space are introduced in subdomains, and L2

integral conditions on the boundaries of subregions are imposed. Then we propose a Neumann-Neumann type method for solving the

resulting system of linear equations. The coarse space problem is solved independently of the local problems and the convergence rate of the method is quasi-optimal i.e. the number of CG iterations grows polylogarithmically as the sizes of the meshes decrease.

High Order One-Way Nesting in One and Two Dimensions.

Assaf Mar–Or*

Inter-departmental Program for Applied Mathematics Technion --- Israel Institute of Technology

Haifa 32000, Israel. E-mail: pierrot@technion.ac.il

Dan Givoli

Department of Aerospace Engineering Technion --- Israel Institute of Technology

Haifa 32000, Israel. E-mail: givolid@technion.ac.il

Abstract:

The problem of one-way nesting, in which global coarse-scale information is incorporated into a regional fine-scale solution, is considered. Carpenter's lateral boundary scheme (Quart. J. R. Met. Soc., vol. 108, 717--719, 1982) and its relation to Sommerfeld's absorbing boundary conditions is presented and analyzed in the context of the scalar wave equation. Carpenter's

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boundary scheme is then compared in one dimension with other possible boundary conditions and is shown to yield better results. Next, the shortcomings of Carpenter's lateral boundary scheme are demonstrated in the case of a two dimensional model problem and a new lateral boundary scheme, which is based on the Hagstrom-Warburton family of high-order absorbing boundary conditions (Wave Motion, vol. 39, 327--338, 2004), is presented. This lateral boundary scheme is then extended further and investigated using a two-dimensional model problem with wave-guide geometry.

A Numerically Efficient Scheme for Elastic Immersed

Boundaries

F.Pacull *(1) and M.Garbey (2) (1) Fluorem – France, fpacull@fluorem.com

(2) Department of Computer Science, UH Houston, Garbey@cs.uh.edu Abstract:

This paper focuses on an efficient numerical scheme for immersed elastic boundary. The immersed bodies considered here have a fixed topology but can encompass large deformation. Our technique takes advantage of a ourier representation of the interface in the Immersed Boundary Method. We will present first an implicit temporal scheme that conserves mass by constrained optimization of the Fourier representation. We will discuss second a fast parallel implementation with domain decomposition in Matlab-MPI. Finally we will give some examples with the bubble test case as well as an innovative artificial motion process of cells in fluids.

A dual iterative substructuring method with a penalty term

Chang-Ock Lee and Eun-Hee Park*

Department of Mathematical Sciences, KAIST, Daejeon 305-701, KOREA Speaker: Eun-Hee Park, mfield@kaist.ac.kr

Abstract:

An iterative substructuring method with Lagrange multipliers is considered for the second order elliptic problem, which is a variant of the FETI-DP method. We propose an augmented Lagrangian functional by a penalty term with a positive parameter ή, which measures the jump across the interface. Then, a dual iterative substructuring method is induced from the saddlepoint problem associated with the augmented Lagrangian. Without any preconditioners, it is

shown that the proposed method is numerically scalable in the sense that for a large !, the condition number of the resultant dual problem is bounded by a constant independent of the subdomain size and the mesh size. According to the numerical results, the presented method is superior to both of the FETI-DP method and the preconditioned FETI-DP by the Dirichlet preconditioner

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from many perspectives such as the conditioning of the dual system, the CG iteration number, and the virtual wall clock time.

The Algebraic Optimized RAS Method

Amik St-Cyr 1,*, Martin J. Gander2 1 Institute for Mathematics Applied to Geosciences, National Center for Atmospheric Research, 1850 Table Mesa Drive, Boulder CO, 80305.

2 Universite de Geneve, Section de Mathématiques 2-4 rue du Lièvre, CP 64CH-1211 Genève

Abstract:

In this work it is shown that no a priori knowledge of the underlying PDE, represented by a matrix at the discrete level, is necessary in order to transform a restricted additive Schwarz (RAS) preconditioner to an optimized RAS (ORAS). A discovery algorithm automatically detecting the elliptic nature of the problem is proposed therein. It is also shown how to extract from the subdomain matrices the various metrics required to apply the optimized transmission conditions (originally discovered at the continuous level for various PDEs). A formula representing a first order approximation to the normal derivative is then applied to the subdomain matrices where connectivity was changed with respect to the global problem. Numerical experiments using elliptic problems discretized with Q1-FEM, P1-FEM, FDM and FVM demonstrate the effectiveness of the method and its algebraic nature.

Dune: The Distributed and Unified Numerics Environment

Oliver Sander Freie Universität Berlin Institut für Mathematik II

Arnimallee 6 14195 Berlin, Germany email: sander@mi.fu-berlin.de

Abstract:

Dune is a set of libraries for grid-based numerical computations. Its main feature is the introduction of an abstract interface which separates grid implementations from the algorithms that use them. Applications are written for the interface instead of for a specific grid implementation. Hence grid implementations can be changed at any moment in algorithm development with minimal effort. Several such implementations are available. Some of them are grids specifically written for Dune, others encapsulate existing well-known finite element codes such as Alberta and UG. Due to the use of modern programming techniques the extra abstraction layer comes at very little additional cost.

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Convergence and Implementation of Robin Domain Decomposition Algorithm for contact Problems

Taoufik Sassi* and Mohamed Ipopa

sassi@math.unicaen.ff ipopa@math.unicaen.fr Laboratoire de mathématiques Nicolas Oresme,

LMNO, Université de Caen Bâtiment Science 3,

avenue du Maréchal Juin, 14032, Caen Cedex, France Abstract:

Contact problems take an important place in computational structural mechanics. Many numerical procedures have been proposed in the literature. They are based on standard numerical solvers for the solution of global problem in combination with a special implementation of the nonlinear contact conditions. The numerical treatment of such non classical contact problems leads to very large and ill-conditioned systems.

Domain decomposition methods are a good alternative to overcome this difficulties (see [2] for example).We propose a and study Robin domain decomposition algorithm for solving unilateral problems arising in contact mechanics [1]. We prove the convergence of this algorithm independently of the discretization step h. Numerical realizations which illustrate the performance of the proposed method will be discussed and results of some numerical examples will be shown. [1] Ipopa, M., Sassi, T., Generalization of Lions' nonoverlapping domain decomposition method for contact problems, Lect. Notes comput. Sci. Eng., Srobl, 2008. [2]* Krause, R. H., Wohlmuth, B., *A Dirichlet-Neumann type algorithm for contact problems with friction, Comput. Vis. Sci. 5 (2002), no. 3, 139--148.

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Parallelization of a constrained three-dimensional Maxwell solver

Franck ASSOUS

Bar Ilan University. Department of Mathematics. Ramat Gan, Israel

Jacques SEGRE* CEA-Saclay, DEN/DM2S/SFME, France

Eric SONNENDRUCKER

Institut de Recherche Mathématique Avancée (IRMA) Unité Mixte de Recherche 7501 du CNRS et de l'Université Louis Pasteur de Strasbourg,

France

Abstract:

The numerical solution of very large three-dimensional electromagnetic field problems are challenging for various applications in the industry. In this presentation, we describe a nonoverlapping domain decomposition approach for solving the three-dimensional Maxwell equations on MIMD computers, based on a mixed variational formulation. It is especially well adapted for the resolution of the Vlasov-Maxwell equations, widely used to simulate complex devices like particle injectors or accelerators. This approach has the important property, that it leads to reuse without modification most of an existing sequential code. The continuity at the interfaces are imposed by duality using Lagrange multipliers. Hence, the resulting parallel algorithm requires only to add an external preconditioned Uzawa solver. We present the results of some numerical experiments on a parallel distributed memory machine that show the efficiency of the method in particular for very large (real-life) problems.

Algebraic multigrid applied to the transonic small disturbances equation.

Shlomy Shitrit* and David Sidilkover

Propulsion Physics Division, Soreq NRC, Israel Abstract:

Capability of computing efficiently compressible potential flow can still be of a significant practical value, for instance, in aerodynamic design problem. Moreover, such a capability becomes crucial in the context of the factorizable methods, which were introduced a few years ago.

The differential operator under consideration (the so-called full-potential operators) is of elliptic type in subsonic flow and its anisotropy becomes more significant as the flow velocity increases and approaches the speed of sound. In the case the flow turns supersonic, the type of the operator changes to hyperbolic.

The algebraic multigrid approach was shown to be very efficient for elliptic problems, both isotropic and anisotropic. Therefore, they would be the natural

31

choice for the purpose of treating the subsonic/sonic flows. However, their extension to the supersonic flow regime appeared to be a difficult problem. First, we had to design a stable relaxation procedure that is efficient and fits the framework of the algebraic multigrid. We required such a relaxation to be

--pointwise (not a block relaxation of any kind); -- direction free (not depending on any particular ordering of the points); -- easily parallelizable. The basic discretization we adopted is based on the original Murman & Cole

idea. We shall describe the whole relaxation scheme together with our extension of the algebraic multigrid method to the supersonic flow regime.

We shall demonstrate the performance of the algebraic multigrid method based on the developed relaxation on a variety of test-cases concerning the variety of flow regimes starting from the low speed and up to supersonic.

Numerical Solutions of an Interface Problem in a Heat Conduction Process Using Some Nonlinear Solvers

Antony Siahaan*1, Choi-Hong Lai2, Koulis Pericleoous3

School of Computing & Mathematical Sciences, University of Greenwich, Old Royal Naval College, Park Row, London SE10 9LS, UK

E-mail: a.l.siahaan@gre.ac.uk, c.h.lai@gre.ac.uk, k.pericleous@gre.ac.uk Abstract:

A Nonoverlapping Domain Decomposition based on the defect method is developed here. We test these methods on some nonlinear heat conduction processes taking place in a multi-chip module which has several geometrically structured subdomains. The arising interface problem is taken care of by imposing a defect equation along subdomain interfaces where the choice of defect equation will be significant in the solution accuracy. The enforcement of the defect equation leads the problem into a system of nonlinear equations which are then solved iteratively by means of some methods in the class of Quasi-Newton and Nonlinear Conjugate Gradient. These nonlinear solvers are developed such that the direct computation of Jacobian matrix can be avoided, thus giving a lighter computation. The performance of these nonlinear solvers will be compared and some issues concerning the development will be discussed.

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Compressible flow equations: decompositions, auxiliary variables and fast solvers

David Sidilkover

Propulsion Physics Division, Soreq NRC, Israel Abstract:

Factorizable methods reflect the mixed character of the fluid flow equations and, therefore, allow to distinguish between the differential operators of different types present in the original system at the discrete level in the same way as at the PDE level. This property makes it possible to address each of the operators can be utilized to design highly efficient solvers by treating each of the operators (co-factors) in the most efficient way, thus leading to a highly efficient solver for the entire system.

The realization of this possibility relies on a certain judiciously chosen ``decomposition'' of the (locally linearized) originalsystem - a variable substitution, that ``separates'' the operators. This decomposition is used then to guide the design of a distributive relaxation . This variable substitution should be as fundamental and general as possible.

Our ``auxiliary variables of choice'' are the classical vorticity—stream function and potential. However, a simple analysis shows that this set of variables is still insufficient for our purpose: it provides no mechanism to drive to zero a certain part of momentum equations residual vector field. This difficulty can be removed by augmenting the classical set of the auxiliary variables by yet another vector, which was given a provisional name ``dual velocity''. This newly ``augmented'' set of variables leads also to some interesting formalism, which will be presented

The solving of boundary value problems by domain

decomposition method without intersection on rectangular quasistructured grids

V.M. Sveshnikov

Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk

State University, Novosibirsk, Russia E-mail: victor@lapasrv.sscc.ru

Abstract:

The variant of domain decomposition method for the solving of boundary value problems on rectangular quasistructured grids, which combine simplicity of rectangular grids with adaptive properties of quasistructured grids is offered. The boundary value problem is formulated as a problem of the solving of the operational equations containing a difference of normal derivatives on interface boundary of subdomains. The numerical algorithm contains three stages. On first of them on a grid, which determined on interface boundary, the system of the algebraic equations concerning values

33

of required function in the grid nodes is constructed. For this purpose a series of auxiliary problems in subdomains is solved. The second stage of algorithm is the solving of the constructed system. At the third stage a final solution of the boundary value problem in subdomains is found. The suggested algorithm is a direct method of decomposition in the sense of it has no iterations on subdomains. Besides it is parallel and consequently is applicable for the solving of boundary value problems on multiprocessing supercomputers. Estimations of the algorithm efficient and the results of numerical experiments are given.

New conditions for non-stagnation of GMRES, and corresponding convergence bounds

Daniel B. Szyld

Temple University Philadelphia, USA

http://www.math.temple.edu/szyld szyld@temple.edu

Abstract:

A well-established condition guaranteeing that GMRES makes some progress, i.e., that it does not stagnate, is that the symmetric part of the coefficient matrix, (A + AT )/2, be positive definite [Elman, 1982]. This condition results in a bound of the convergence rate for the iterative method which depends on the minimum eigenvalue of (A + AT )/2 and of the norm of A. This bound is usually very pessimistic. Nevertheless, it has been extensively used by the DD community to show that preconditioned problems have a convergence bound for GMRES which is independent of the underlying mesh size of the discretized partial differential equation. In this talk we discuss new and more general conditions on the coefficient matrix so that one can guarantee that there is no stagnation of GMRES. These conditions do not require the symmetric part of the coefficient matrix to be positive definite. Thus, we enlarge the class of matrices for which a bound of the convergence rate for GMRES is available. We present several examples for which the new conditions are satisfied, while the Elman bound is not. It is hoped that the new bounds can be used to show mesh independence of preconditioners for which the Elman bound is not applicable, and similarly to encourage development of new optimal or nearly optimal preconditioners. (joint work with Valeria Simonicini, Universitá di Bologna)

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Risolv - Robust Iterative Solver

Hillel Tal-Ezer Academic College of Tel-Aviv Yaffo

Abstract:

Domain decomposition is a popular approach for designing preconditioning algorithm. In order to use it for solving large linear system Ax = b one need also an adequate accelerator. Gmres, for example, is such an algorithm which is used in cases where A is general, non Hermitian matrix. Unfortunately, it suffers from lack of robustness. The algorithm can exhibit very slow rate of convergence or complete stagnation. In this talk we would like to present a new accelerator, named Risolv, which overcomes this drawback. Whenever a theoretical polynomial (in the preconditioned matrix) algorithm can solve the linear system, so does Risolv. The algorithm is especially efficient in cases where there is a need to solve many systems which share the same matrix and differ by the right hand side vectors. In this case, implementing Risolv for most of the systems is almost free of inner-products. This feature can result in significant saving of cpu, specially in parallel computing

A domain decomposition method for discontinuous Galerkin discretizations of Helmholtz problems with Lagrange

multipliers

Charbel Farhat, Radek Tezaur, and Jari Toivanen* Institute for Computational and Mathematical Engineering

Stanford University Stanford, CA 94305, USA

Abstract:

A nonoverlapping domain decomposition method is described for Helmholtz problems discretized by a discontinuous Galerkin finite element method. The discretization uses plane wave basis functions and Lagrange multipliers to enforce a weak continuity of solution over element interfaces. A system of linear equations is formulated for the Lagrange multipliers on the subdomain interfaces. This poorly conditioned system is solved iteratively with a local preconditioner after it has been projected onto the complement of a coarse space in the same way as in the FETI-H method. Numerical experiments study the iterative solution of two-dimensional and three-dimensional model problems and compare the convergence and accuracy to the FETI-DPH method.

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AN EFFICIENT APPROACH FOR UPSCALING PROPERTIES OFCOMPOSITE MATERIALS WITH HIGH CONTRAST OF

COEFFICIENTS

R. EWING1, O. ILIEV2,6, R. LAZAROV3, I. RYBAK4, AND J. WILLEMS5* Abstract: An efficient approach for calculating the effective heat conductivity for a class of industrial composite materials, such as metal foams, fibrous glass materials, and the like, is discussed. These materials, used in insulation or in advanced heat exchangers, are characterized by a low volume fraction of the highly conductive material (glass or metal) having a complex, network-like structure and by a large volume fraction of the insulator (air). We assume that the composite materials have constant macroscopic thermal conductivity tensors, which in principle can be obtained by standard up-scaling techniques, that use the concept of representative elementary volumes (REV), i.e. the effective heat conductivities of composite media can be computed by post-processing the solutions of some special cell problems for REVs. We propose, theoretically justify, and numerically study an efficient approach for calculating the effective conductivity for media for which the ratio δ of low and high conductivities satisfies δ << 1. We start from the known Domain Decomposition approaches for such problems, separating the subdomains in the way that each subdomain contain only one material. We show that for our purposes, one essentially only needs to solve the heat equation in the region occupied by the highly conductive media. For the considered class of problems, we show that under certain conditions on the microscale geometry, the proposed approach produces an upscaled conductivity that is O(δ) close to the exact upscaled permeability. A number of numerical experiments are presented in order to illustrate the accuracy and the limitations of the proposed method. Applicability of the presented approach to upscaling other similar problems, e.g. flow in fractured porous media, is also discussed. Keywords: effective heat conductivity, permeability of fractured porous media, numerical upscaling, fibrous insulation materials, metal foams. 1 Institute for Scientific Computation, Texas A&M University College Station, TX, 77843, USA, richard-ewing@tamu.edu 2 Fraunhofer Institut f¨ur Techno- undWirtschaftsmathematik, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany, iliev@itwm.fhg.de 3Department of Mathematics Texas A&M University College Station, TX, 77843,USA, lazarov@math.tamu.edu 4Institute of Mathematics, National Academy of Sciences of Belarus, Surganov Str. 11, 220072 Minsk, Belarus, rybak@im.bas-net.by 5Fraunhofer Institut f¨ur Techno- undWirtschaftsmathematik, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany, willems@itwm.fhg.de 6 Institute of Mathematics, Bulg. Acad.Sci., Acad.G.Bonchev str., bl.8, 1113 Sofia, Bulgaria

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Part IV

Minisymposuim Talks

37

Mini-Symposium 1

Numerical modeling and solution methods for the

multi-domain multi-scale models

Organizer: Fraçois-Xavier Roux Speakers:

• Hachmi Bendhia

• Christian Rey

• Frédéric Feyel,

• David Dureissex

• François-Xavier Roux

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The Arlequin methodology

Hachmi Ben Dhia LMSSMat, Ecole Centrale Paris, UMR 8579

Abstract:

The Arlequin method [1], [2] is a general numerical modeling methodology that superposes models and glues them to each others while partitioning the enrgies. Thus, by construction, this approach allows for local multimodel and multiscale analyses of existant global coarse numerical models. This multiscale methodology will be explained, analyzed [3], [4] and exemplified (e.g. [5]). As a matter of fact, the Arlequin method which is actually a local multimodel partitioning framework leads to discrete problems that are rather similar to the ones obtained when using Domain Decomposition Methods to solve monomodel problems derived from PDE's. Thus one can take advantage of the development made in the latter field to solve the discrete Lagrangian or augmented lagrangian Arlequin problems as done in [6]. References: [1] H. Ben Dhia, Multiscale mechanical problems : the Arlequin method, Comptes Rendus de l'Acadיmie des Sciences, Sיrie IIb, 326, (1998) 899-904. [2] H. Ben Dhia, Numerical modeling of multiscale problems: the Arlequin method, in Proceedings of the First European Conferenceon Computational Mechanics, Muenchen, (1999). [3] H. Ben Dhia, G. Rateau, Mathematical analysis of the mixed Arlequin method, Comptes Rendus de l'Acadיmie des Sciences Paris Sיrie I, 332, (2001) 649-654. [4] H. Ben Dhia, Global-Local approaches: the Arlequin framework, European Journal of Computational Mechanics, 15, (1-3), (2006) 67-80. [5] H. Ben-Dhia, G. Rateau, The Arlequin method as a flexible engineering design tool, International Journal for Numerical Methods in Engineering, 62, (2005) 1442-1462. [6] H. Ben Dhia, N. Elkhodja, F-X Roux, Multimodeling of multi-alterated structures in the Arlequin framework. Solution with a Domain-Decomposition solver (selected in European Journal of Computational Mechanics, 2007)

Domain decomposition and non linear relocalization with or without overlapping

Chritian Rey, Julien Pebrel, Pierre Gosselet

Ecole Normale Supérieure de Cachan christian.rey@lmt.ens-cachan.fr

Abstract:

We consider the evolution of large structure undergoing localized nonlinear phenomenon (plasticity, damage, cracking, microbuckling …). In that case, even if most of the structure remains elastic, the convergence of the nonlinear solver (typically Newton-Raphson algorithm) is driven (and then slowed) by the local phenomenon. We propose a strategy intending to solve the nonlinearity at a local scale. The method relies on a domain decomposition introduced by a partition of the unity applied to the energy of the system. The

39

connection is insured using augmented Lagrange multipliers, so that the method can be interpreted as finding ''good'' mixed conditions on the interface.

Then a classical algorithm (FETI, BDD) is employed, with local treatment of the non-linearity: in order to find the exact interface conditions, the nonlinear problem is condensed on the interface, then it is linearized leading to the resolution of a succession of linear interface problems (global) and nonlinear subdomain problems (local). In our presentation, the method will be first exposed and related to existing strategies, then assessed on academic problems (and if possible more complex situations).

FE² and similar methods: overview and link with domain decomposition

Frédéric Feyel

ONERA – MMSD frederic.feyel@onera.fr

Abstract:

The FE² term has been introduced some years ago to designate multiscale models where the mechanical behavior of the representative volume element is computed, when needed, using a direct finite element computation over a unit cell.

Such models lead to a numerical framework in which the macroscopic behavior is computed as usual using the finite element method. During this macroscopic computation when one has to compute the material response, an embedded finite element computation is done over the unit cell. It thus results in simultaneous finite element computations at both scales. Domain decomposition methods are easily used at the macroscale to speed-up the computation.

Some interesting questions have been raised about the multiscale status of these FE² methods, and more generally about what are multiscale methods, from both a computational point of view and a mechanical point of view. Some insights will be given during the talk around this question.

The talk will be organized as a travel based on the size of the material heterogeneities. For each order of magnitude of this representative material size, we will try to propose methods able to compute heterogenous structures for each zone, including FE² models.

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A multiscale domain decomposition for the simulation of a non smooth structure, involving a numerical homogenization

Damien Iceta, Pierre Alart and David Dureisseix Laboratoire de M´ecanique et G´enie Civil (LMGC)

University Montpellier 2 / CNRS UMR5508 Place E. Bataillon, F-34095 MONTPELLIER CEDEX 5, FRANCE

{iceta,alart,dureisseix}@lmgc.univ-montp2.fr

Abstract:

The simulation of a granular media behavior, with contact and friction be- tween grains, is a typical example of a non smooth mechanical system [8]. A

second example, for which no reconfiguration of connectivity occurs due to large displacements, is the dynamical evolution of a tensegrity structure [9]. This is and innovative civil engineering structure, composed with bars and cables. Non smoothness is diffuse over the whole structure, due to possible slackness of cables under overloading [10].

When a large scale structure is considered, a domain decomposition is used to solve the problem. Since the constitutive behavior is a non univoque and non smooth relation between primal and dual quantities, a mixed approach is used: both forces and displacements are the unknowns [5, 3]. With the proposed approach, non smoothness is localized within the subdomains, while the interfaces possess a perfect behavior. This choice is similar to the one used in [2], and somehow the dual of the one used in [7] where non linearities are isolated on the interfaces.

The splitting of mechanical fields arising on interfaces (forces and displace-ments) into a macroscopic part and a microscopic complement allows to build a multiscale approach, for which the relationship between macro forces and dis-placements leads to an homogenized behavior of the subdomains [6]. Thesefields constitute the coarse space of the multilevel domain decomposition [1].

The solver should deal with both the primal and dual unknowns. At least two different solvers can be used: (i) a dedicated version of the Large Time INcrement approach [5], similar to an augmented Lagrangian approach, and (ii) a derived version of the Non Smooth Contact Dynamics approach [4, 8], similar to a non smooth Jacobi/Gauss-Seidel solver. References: [1] P. Alart and D. Dureisseix. A scalable multiscale LATIN method adapted to nonsmooth discrete media. Computer Methods in Applied Mechanics and Engineering, 197(5):319–331, 2008.1 [2] M. Barboteu, P. Alart, and M. Vidrascu. A domain decomposition strategy for nonclassical frictional multi-contact problems. Computer Methods in Applied Mechanics and Engineering, 190:4785–4803, 2001. [3] L. Champaney and D. Dureisseix. A mixed domain decomposition ap-proach. In F. Magoul`es, editor, Mesh Partitioning Techniques and Do- main Decomposition Methods. Civil-Comp Press / Saxe-Coburg Publica- tions, 2007. To appear. [4] M. Jean. The non-smooth contact dynamics method. Computer Methods in Applied Mechanics and Engineering, 177:235–257, 1999.

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[5] P. Ladev`eze. Nonlinear Computational Structural Mechanics — New Ap- proaches and Non-incremental Methods of Calculation. Springer Verlag, 1999. [6] P. Ladev`eze and D. Dureisseix. A micro / macro approach for parallel computing of heterogeneous structures. International Journal for Compu-tational Civil and Structural Engineering, 1:18–28, 2000. [7] P. Ladev`eze, A. Nouy, and O. Loiseau. A multiscale computational ap-proach for contact problems. Computer Methods in Applied Mechanics and Engineering, 191:4869–4891, 2002. [8] J. J. Moreau. Numerical aspects of sweeping process. Computer Methods in Applied Mechanics and Engineering, 177:329–349, 1999. [9] R. Motro. Tensegrity. Hermes Science Publishing, London, 2003. [10] S. Nineb, P. Alart, and D. Dureisseix. Domain decomposition approach for nonsmooth discrete problems, example of a tensegrity structure. Computers and Structures, 85(9):499–511, 2007.

Non conforming FETI-2LM method

François-Xavier Roux ONERA and University of Paris 6

roux@onera.fr Abstract:

For heterogeneous composite materials, it may appear relevant to use non conforming meshes in order to be able to have highly refined regions in the mesh.

In such a context, the FETI-2LM method, based on Robin interface matching conditions, that is very efficient for highly heterogeneous subdomains, appears to be the solution method of choice. In this paper, we present the extension of the FETI-2LM method using a recently introduced generalization of the mortar method for non conforming meshes.

This method will be compared to another approach using also the FETI-2LM method, consisting in localizing the non conforming interfaces by putting both sides of the interface inside one subdomain.

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Mini-Symposium 2a and 2b

DE-Constrained

Optimization Organizers:

• Hoppe • Kornhuber • Krause

Speakers:

• Grease • Gross • Hintermuller • Hinze • Schiela • Tai • Weiser • Xu • Zulehner • +2 of the organizers

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Global convergence of nonsmooth Newton methods for constrained minimization

Carsten Gräser

Freie Universität Berlin Abstract:

Constrained minimization problems with inequality constraints can be formulated as nonsmooth equations in various ways. Nonsmooth Newton methods turned out to be an efficient approach for the solution of such problems. Unfortunately the particular algorithms converge only locally and rely on exact solution of the subproblems in general. Common approachs try to overcome this by the construction of artificial merit functions.

We show that natural problem inherent energies can be used as merit functions if the nonsmooth nonlinear strategy is chosen appropriately. The presented approachs lead to global convergence even in the case of very inexact solution of the linear subproblems. A Multiscale Approach for Convex and Non-convex Constrained Minimization.

C. Gross, R. Krause Institute for Numerical Simulation, University of Bonn

Abstract:

In this talk, we present a solution strategy for convex and non-convex minimization problems subject to pointwise constraints. This strategy is based, on the one hand, on a non-linear multilevel algorithm and, on the other, on a trust-region strategy to solve the occuring non-linear minimization problems. Hence, the function is minimized iteratively by generating a sequence of admissible corrections. By imposing only slightly restrictive assumptions on the function and on the corrections, we are able to prove convergence to first- and second-order critical points, as well as, local quadratic convergence. We furthermore illustrate the effectiveness of our implementation of this algorithm by presenting results from three-dimensional non-linear elasticity.

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A fully practical semi-smooth Newton method in variational-

discrete pde constrained optimization

Michael Hinze (Joint work with Morten Vierling)

Hamburg

Abstract: We consider a semi-smooth Newton algorithm for variational discretizations of control constrained elliptic and parabolic optimal control problems. In this approach only the solution operator associated with the pde is discretized while the control itself is not. For this discrete approach we present a semismooth Newton method which operates in function space but is fully implementable as its iterates can be represented on refinements of the original triangulation allowing jumps along the border between active and inactive set. We prove fast local convergence of the algorithm and propose a structure exploiting globalization strategy. Numerical tests for elliptic and parabolic optimal control problems with box constraints on the control confirm our analytical findings. We further present numerical results for state constrained elliptic optimal control problems based on the Lavrentiev relaxation.

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Interior Point Methods in Function Space for State Constrained Optimal Control

Anton Schiela

ZIB Berlin

Abstract:

We consider a class of interior point methods that can be applied to PDE constrained optimization problems with state constraints. As a distinguishing feature, these methods are constructed and analysed in function space. This approach makes it easier to exploit the analytical structure of the optimal control problem. Moreover, implementations are solely based on quantities that are well defined in function space.

We study structure and convergence of homotopy paths generated by logarithmic and rational barrier functions. It turns out that a proper type of barrier function has to be chosen in order to guaranty existence and strict feasibility of solutions in function space.

For the right type of barrier functions, the convergence of corresponding primal Newton path-following schemes to the solution of the original state constrained problem can be established.

We propose, as an algorithmic modification, a pointwise damping strategy, which retains the theoretical properties of the pathfollowing scheme, but leads to significantly faster convergence in practice, compared to the pure primal Newton corrector. We conclude with numerical experiments that show the efficiency of this method.

Hyperthermia Treatment Planning with Interior Point Methods

Martin Weiser ZIB, Berlin

Abstract: The talk will review the problem of designing an optimal cancer treatment based on heating large tumors by microwave radiation. Modelling of heat distribution inside the human tissue is addressed. Specific aspects of the problem with strong impact on interior point methods used for the solution of the arising PDE-constrained semi-infinite program are discussed and illustrated with numerical examples.

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Solving A Saddle Point Problem by Solving One Related Symmetric Positive Definite System

Jinchao Xu PennState

Abstract: In this talk, I will first explain how the solution of a saddle point problem can be reduced to that of a nearly singular symmetric positive definite system. I will then explain how these nearly singular systems can be solved in an optimal and robust fashion by using properly designed iterative procedure based on the method of subspace corrections.

Patch smoothers for saddle point problems with applications to PDE constrained optimization problems

Renè Simon,

SFB 013, Johannes Kepler University Linz, Altenbergerstrasse 69, A-4040 Linz, Austria;

e-mail: rene.simon@sfb013.uni-linz.ac.at

Walter Zulehner*, Institute of Computational Mathematics, Johannes Kepler University,

Altenbergerstrasse 69, A-4040 Linz, Austria; e-mail: zulehner@numa.uni-linz.ac.at

Abstract: In this talk we consider a multigrid method for solving the discretized optimality system (Karush-Kuhn-Tucker system, in short: KKT system) of a PDE-constrained optimization problem. In particular, we discuss the construction of an additive Schwarz-type smoother for a certain class of elliptic optimal control problems. Each iteration step of the additive Schwarz method requires the solutions of several small local saddle point problems. Strategies for constructing the local problems are presented, which allow a rigorous multigrid convergence analysis.

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Mini-Symposium 3

Computational and Programming techniques for

Domain Decomposition.

Organizer: O. Pironneau.

Speakers:

• Pascal Have • Frederic Hecht • Alexei Lozinski • Frederic Nataf

48

Some Domain decomposition methods in freefem++

Frédéric Hecht Olivier Pironneau * Universit_e Pierre et Marie Curie-Paris6, UMR 7598 Laboratoire Jacques-

Louis Lions, Paris, F-75005 France I will present an example of domain decomposition technique, one in in sequential and in parallel framework, the implementation of the discontinuous enrichment method (DEM) [3] with mesh adaptation. First FreeFem++ is a free software to solve bidimensional PDE's and the characteristics of FreeFem++ are: • Problem description (real or complex valued) by their variational

formulations, with access to the internal vectors and matrices if needed. • Multi-variables, multi-equations, bidimensional (or 3D axisymmetric) ,

static or time depen-dent, linear or nonlinear coupled systems; however the user is required to describe the iterative procedures which reduce the problem to a set of linear problems.

• Easy geometric input by analytic description of boundaries by pieces; however this software is not a CAD system; for instance when two boundaries intersect, the user must specify the intersection points.

• Automatic mesh generator, based on the Delaunay-Voronoi algorithm. Inner point density is proportional to the density of points on the boundary [?].

• Metric-based anisotropic mesh adaptation. The metric can be computed automatically from the Hessian of any FreeFem++ function [1].

• High level user friendly typed input language with an algebra of analytic and _nite element functions.

• Multiple finite element meshes within one application with automatic interpolation of data on different meshes and possible storage of the interpolation matrices.

• A large variety of triangular finite elements : linear and quadratic Lagrangian elements, dis-continuous P1 and Raviart-Thomas elements, elements of a non-scalar type, mini-element,. . . (but no quadrangles).

• Tools to define discontinuous Galerkin formulations via finite elements P0, P1dc, P2dc and

• keywords: jump, mean, intalledges. • A large variety of linear direct and iterative solvers (LU, Cholesky, Crout,

CG, GMRES, UMFPACK) and eigenvalue and eigenvector solvers. • Near optimal execution speed (compared with compiled C++

implementations programmed directly). • Online graphics, generation of ,.txt,.eps,.gnu, mesh _les for further

manipulations of • input and output data. • Many examples and tutorials: elliptic, parabolic and hyperbolic problems,

Navier-Stokes flows, elasticity, Fluid structure interactions, Schwarz's domain decomposition method, eigen-value problem, residual error indicator...

• An parallel version using mpi

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1 The DEM method to solve a Poisson Problem The Discrete Enrichment Method of Charbel Farhat et al solves

( ) 0|, =!"=#$#% &ufuA by

HhVwubwfwbwua !"#$==+ µµ% ,0),(),,),(),( where, !=U k

E being the lines of discontinuity of w ,

[ ]!! ""#

$=%%= wwbuAWwua

T ),(,),(

Let HT be a triangulation of where ! is the approximate length of the edges

E of the triangles T , and let KE be the internal edges of

HT . Each triangle T

of HT is divided into smaller triangles of approximate size h independently.

On each T if the value of u at the 3 vertices are known as well as its integrals on the 3 edges, then the values on the vertices of the small triangles can be found by a local problem. This is because, for any continuous piecewise linear U on

HT and any !"# there is a unique solution u which is zero the vertices

of HT and has mean zero on the edges E and is solution of

wwbwUawfwua !"+= ),(),(),(),( # Continuous piecewise linear onT , zero on T!

Furthermore this problem decomposes in N independent problems on each of the N triangles of

HT .

The interpolation space hV is made of functions which are continuous at all the

vertices of HT and inside each T of

HT but discontinuous at the edgesE .

However we require that the mean jump [ ] 0=!E u on each edge of HT and this

constraint is enforced by a Lagrange multiplier ! constant on E . The set of piecewise polynomial functions of degre k on the edges E is called

H! .

I will show how to implement this algorithms in freefem++ in sequential, in parallel,and a how to do mesh adaptation on each finite mesh of triangle

HTT ! .

References: [1] F. Hecht The mesh adapting software: bamg. INRIA report 1998. http://www-rocq.inria.fr/gamma/cdrom/www/bamg/eng.htm. [2] F. Hecht, K. Ohtsuka, and O. Pironneau. FreeFem++ manual. Universite Pierre et Marie Curie, 2002{2005. on the web at http://www.freefem.org/ff++/index.htm. [3] Charbel Farhat, Isaac Hararib, and Leopoldo P. Franca The discontinuous enrich-ment method Computer Methods in Applied Mechanics and Engineering Volume 190, Issue 48, 28 September 2001, Pages 6455-6479

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Harmonic Finite Element Refinments

Alexei Lozinski Institut de Mathématiques de Toulouse, Université Paul Sabatier

31062 Toulouse, France Abstract:

We discuss a method for the numerical solution of elliptic problems with multi-scale data using multiple levels of not necessarily nested grids. The first version of the method was studied in [1]. It consists in calculating successive corrections to the solution in small sub-domains (patches) whose fine triangulations are not necessarily conforming to the coarse triangulation of the whole domain. While this method works fine in the case of nested triangulations, its convergence can be very slow if the fine grid is slightly shifted with respect to the coarse grid. We present here a new variant of the method (see [2]) whose convergence rate is essentially independent of the mutual placement of the two grids and which turns out to be significantly faster than the original one. The novelty of the method consists in restricting finite element functions on the coarse grid to be approximately harmonic inside the sub-domain where a finer triangulation is applied. We will discuss also alternative implementations of these methods that further reveal their connections with Schwarz domain decomposition algorithms without overlapping.

1. R. Glowinski, J. He, A. Lozinski, J. Rappaz and J. Wagner, Finite element approximation of multi-scale elliptic problems using patches of elements, Numer. Math. (2005) 101(4), 663 – 687.

2. J. He, A. Lozinski and J. Rappaz, Accelerating the method of finite element patches using approximately harmonic functions, Comptes rendus Mathematique (2007) 345(2), 107 – 112.

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Mini-Symposium 4

Organizer: Laayouni-Gander

Speakers:

• Berninger • Dubois • Gander • Japhet • Laayouni/Gander • Picasso • Amik St. Cyr

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Non-overlapping domain decomposition for the Richards equation via Nemytskij operators

Heiko Berninger*, Ralf Kornhuber and Oliver Sander

Freie Universität Berlin Abstract:

We present new results on transmission problems related to the Richards equation in heterogeneous porous media. The Richards equation, which describes saturated-unsaturated groundwater flow, is discretized in time with an explicit treatment of the gravitational term. Then, different Kirchhoff transformations on the subdomains containing different soil-types lead to a coupling of local convex minimization problems across the interfaces. The nonlinear coupling is provided by Nemytskij operators acting on the trace space. The corresponding transmission conditions give rise to nonlinear Dirichlet--Neumann or Robin methods for which convergence results have been obtained in one space dimension ([1], [2]). We solve the local problems efficiently and robustly by monotone multigrid [3]. For the domain decomposition iterations, too, no further linearization is applied.

Our numerical results provide a detailed comparison of the Dirichlet-Neumann method and the Robin method for problems related to the stationary Richards equation in 2D. Furthermore, we present a numerical example in 2D wherein we apply the Robin method to the Richards equation in four different soils with surface water and realistic hydrological data. References: [1] H.~Berninger. {\em Domain Decomposition Methods for Elliptic Problems with Jumping Nonlinearities and Application to the Richards Equation.} Dissertation, FU Berlin, October 2007. [2] H.~Berninger, R.~Kornhuber and O.~Sander. On nonlinear {D}irichlet--{N}eumann algorithms for jumping nonlinearities. In: O.B.~Widlund and D.E.~Keyes, editors, {\em Domain Decomposition Methods in Science and Engineering~XVI}, pp.\ 483--490, Springer, 2007. [3] R.~Kornhuber. On constrained Newton linearization and multigrid for variational inequalities. {\em Numer.\ Math.}, 91:699--721, 2002.

Behavior of optimized Schwarz methods for multiple subdomains and coarse space corrections.

Olivier Dubois

Institute for Mathematics and its Applications, University of Minnesota

Abstract:

Optimized transmission conditions are known to greatly improve the convergence of the Schwarz iteration. In the derivation of these transmission conditions, the convergence factor is optimized for a model problem with two subdomains. In this work, we study optimized Schwarz methods with Robin transmission conditions when applied to multiple subdomains. We experiment

53

with several choices of coarse space corrections, and compare the performance with the classical two-level additive Schwarz preconditioner.

We demonstrate in particular that, given a method for coarse space correction, the best parameters for the two-level Schwarz iteration may be very different from the best parameters for the one-level Schwarz iteration. Numerical experiments in one and two dimensions will be presented.

Discontinuous Galerkin and nonconforming in time optimized

Schwarz waveform relaxation for coupling heterogeneous problems

Laurence Halpern, Caroline Japhet*

Laboratoire d’Analyse, G´eom´etrie et Applications, Universit´e Paris XIII, Avenue J-B Cl´ement, 93430 Villetaneuse, France

Jérémie Szeftel

Department of Mathematics, Princeton University, and C.N.R.S., Mathématiques Appliquées de Bordeaux, Université Bordeaux

Abstract:

In the context of long time computations in highly discontinuous media such as far field simulations of underground nuclear waste disposal or in climate modeling, it is of importance to split the computation into subproblems for which robust and fast solvers can be used. Optimized Schwarz waveform relaxation algorithms have been described for linear convection-diffusion-reaction problems in [1], and provide an efficient approach even with rotating velocity fields [4]. These algorithms are global in time, and thus allow for the use of non conforming space-time discretizations. They are therefore well adapted to coupling models with very different spatial and time scales, as in ocean-atmosphere coupling or nuclear waste disposal simulations [3]. Our final objective is to propose efficient algorithms with a high degree of accuracy, for heterogeneous advection-diffusion problems with discontinuous coefficients. The goal is to split the time interval into time windows, and to perform, in each window, a small number of iterations of an optimized Schwarz waveform relaxation algorithm, using second order optimized transmission conditions . The subdomain solver is the discontinuous Galerkin method in time, and classical finite elements in space. The coupling between the subdomains in done by a simple projection procedure, where no composite grid on the boundary is needed. This approach has been introduced in [2], with promising one dimensional numerical results, and we extend here the analysis in higher dimension. The nonconforming domain decomposition method is proved to be well posed, and the iterative solver to converge for simple geometries. We present two-dimensional numerical results to illustrate the performances of the method. References:

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1. D. Bennequin, M J. Gander, and L. Halpern A Homographic Best Approximation Problem with Application to Optimized Schwarz Waveform Relaxation, To appear in Math. of Comp. 2. E. Blayo, L. Halpern and C. Japhet. Optimized Schwarz waveform relaxation algorithms with nonconforming time discretization for coupling convection-diffusion problems with discontinuous coefficients. Domain Decomposition Methods in Science and Engineering XVI 3. M. Gander, L. Halpern and M. Kern. A Schwarz Waveform Relaxation Method for Advection–Diffusion–Reaction Problems with Discontinuous Coefficients and non- Matching Grids. Domain Decomposition Methods in Science and Engineering XVI 4. V. Martin, An optimized Schwarz waveform relaxation method for the unsteady convection diffusion equation in two dimensions, Appl. Numer. Math. 52 (2005), no. 4, 401–428.

From Schwarz's idea to a New Era of Optimized Domain Decomposition Methods

Laayouni-Gander

Abstract:

The classical Schwarz algorithm is one of the very well known iterative schemes for solving partial differential equations. The original idea was introduced by Schwarz in 1870 to prove existence and uniqueness of solution of poisson's problem on irregular domains. The advances in computer performance and multiprocessor architectures have permitted the Schwarz method to become one of the popular iterative algorithms amongst other solution methods.Different investigations have been devoted to improving the slow convergence of the classical Schwarz method, in particular for nonoverlapping decompositions. A remedy and a novel idea was to modify the transmission conditions, thus changing the information that is communicated between the subdomains. A new class of domain decomposition methods was then introduced, known as optimal and optimized domain decomposition methods.Although computer architectures have evolved significantly, we still need efficient and optimal iterative algorithms. Optimized Schwarz methods have shown to be efficient in solving several differential models.

This minisymposium will focus on the evolution progress of the original idea of Schwarz to the optimized Schwarz era. The minisymposium will also consider some technical issues and challenges, including the extension of Optimized Schwarz methods to more complex systems, the treatment of periodic problems, the design of coarse space corrections for multiple subdomains, and the convergence analysis for more complicated geometries.

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Finite elements with patches

Marco Picasso*, Jacques Rappaz, Vittora Rezzonico

Ecole Polytechnique Fédérale de Lausanne

Abstract: We develop a discretization and solution technique for elliptic problems whose solutions may present strong variations, singularities, boundary layers and oscillations in localized regions. We start with a coarse finite element discretization with a mesh size $H$, and we superpose to it local patches of finite elements with finer mesh size $h<<H$ to capture local behaviour of the solution. The two meshes (coarse and patch) are not necessarily compatible. The corresponding linear system is solved using a subspace correction method. Similarly to mesh adaptation methods, the location of the fine patches is identified by an a posteriori error estimator. Unlike mesh adaptation, no remeshing is involved. We discuss the implementation and illustrate the method on academic and industrial examples.

Optimized Schwarz preconditioning for spectral elements based magnetohydrodynamics.

Amik St. Cyr

Institute for Mathematics Applied to Geosciences, National Center for Atmospheric Research

Duane Rosenberg

Institute for Mathematics Applied to Geosciences, National Center for Atmospheric Research

Sang Dong Kim

Department of Mathematics, Kyungpook National University

Abstract: A recent theoretical result on optimized Schwarz algorithm presented in [SISC, 29(6),pp 2402–2425] enables the modification of an existing Schwarz procedure to its optimized counterpart. In this work, it is shown how to modify a bilinear FEM based Schwarz preconditioning strategy to its optimized version. The latter is then employed to precondition the pseudo Laplacian operator arising from the spectral element discretization of the magnetohydrodynamic equations. In order to yield resolution independence in the Krylov iteration count various experiments are performed with a coarse solver based on radial basis functions.

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Mini-Symposium 5

Milestones in the Development of Domain Decomposition Methods: a Historical Perspective

Organizer:

• Martin J. Gander • David E. Keyes

Speakers:

• Olof Widlund • Petter Bjorstad • Roland Glowinski • Jinchao Xu • David Keyes • Francois-Xavier Roux • Frederic Nataf • Xiao-Chuan Cai • Laurence Halpern

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Coarse Space Components of Domain Decomposition Algorithms

Olof Widlund

Abstact:

An historical overview of the development of some quite exotic coarse spaces, i.e., global low-dimensional components of domain decomposition preconditioners will be given. Some highlights:

The beginning, there was a realization that a second level is required in order to obtain scalability of these iterative methods, i.e., bounds on their convergence rates which are independent of the number of subdomains into which the region of the elliptic problem has been divided. By the time of DD1, a series of four important papers by Bramble, Pasciak, and Schatz had been written; they turned out to be very influential both in terms of algorithm development and new technical tools. With the development of relatively exotic coarse spaces, bounds on the convergence rates could be localized to individual subdomains, which allowed for bounds which are independent of even large jumps in material properties across the interfaces between subdomains.

Soon thereafter, the two-level overlapping Schwarz methods were introduced and the null space property was formulated.

Another important milestone was the introduction of the balancing domain decomposition methods, where the original algorithms soon were augmented by a coarse component. Already in these algorithms, the coarse component is defined in an implicit way and the same is true for the one-level FETI methods and the BDDC and FETI-DP algorithms, which now represent the state of the art.

The coarse component of a domain decomposition method can also serve purposes other than just providing some global interchange of information in each step of the iteration; almost incompressible elasticity will serve as an example. In these cases, the coarse spaces need to be enriched beyond the point when the null space condition is met.

To Overlap or not to Overlap

Petter Bjorstad

Abstract:

In this talk, we provide a historical context and motivation for overlapping as well as non-overlapping domain decomposition algorithms. We will further, by way of a simple example, show some of the relationships between these methods that were discovered quite early. The talk will further discuss some trade-off issues, then end with a look ahead as we (again) need to adopt to massively parallel computer systems.

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On Fictitious Domain Methods

Roland Glowinski Abstract:

During this lecture (complementary to the one dedicated to Moshe Israeli) the author would like to discuss various approaches for the direct numerical simulation of particulate flow and providethe rational for the particularfictitious domain methodhe and his collaborators have been advocating.This presentation will be illustrated by the visualization (including movies)of the results of various numerical experimentsfor 2 and 3-dimensional particulate flow.

On the method of subspace corrections

Jinchao Xu

Abstract

In this talk, an overview will be given on the development of the method of subspace correction based on space decomposition. Main ideas will be explained on its basic algorithmic framework, relevant theoretical analysis and practical applications. Multigrid and domain decomposition methods will be presented as primary examples of this type of methods.

The FETI Method

Francois-Xavier Roux

Abstract:

The FETI method was the first domain decomposition method based on the

use of Lagrange multipliers for enforcing interface continuity condition. With the FETI method, the interface variable is not the trace of the solution, but the Lagrange multiplier that is in the dual space and, so, the local problems are associated with Neumann boundary conditions.

The main specific feature of the FETI method is that it contains a built in "coarse grid" preconditioner. Since the local Neumann problems are ill posed in all subdomains that do not inherit Dirichlet boundary conditions from the global problem, the condensed interface problem of the FETI method is a mixed problem that is solved via a projected conjugate gradient algorithm. The projection phase consists in solving a global coarse problem whose

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unknowns are, in the case of structural analysis, the rigid body motion coefficients of the floating subdomains.

The FETI solution method can be easily extended to the case of non matching grids on the interface, since the introduction of Lagrange multipliers for the weak continuity conditions is the natural way to tackle such problems. With the FETI-2 method, a very general and flexible way to introduce more sophisticated coarse grid preconditioners in the method has been designed.

For indefinite problems, like the Helmholtz equation, the FETI method has been extended to the case of Robin boundary conditions on the interface, with one (FETI-H) or two (FETI-2LM) Lagrange multipliers. The FETI-2LM method has given now a very general methodology to design domain decomposition methods for a wide range of discretizations of PDEs.

A variant of FETI, mixing the dual approach based on the introduction of Lagrange multipliers with the primal Schur complement method, called FETI-DP, has become more and more popular in the past few years. Another mixed variant has also been developed for mixed formulations of EDPs, like uncompressible elasticity.

This paper will present the panorama of what is now the family of FETI methods, and will give an insight of their various features as well as an idea on the criteria for choosing the right method for a given application.

Optimized Schwarz Methods

Frederic Nataf

Abstract:

The strategy of domain decomposition methods is to decompose the computational domain into smaller subdomains. Each subdomain is assigned to one processor. The equations are solved on each subdomain. In order to enforce the matching of local solutions, interface conditions have to be imposed on the boundary between subdomains. These conditions are enforced iteratively. The convergence rate of the associated algorithm is very sensitive to these interface conditions. The Schwarz method is based on the use of Dirichlet conditions. It is slow and requires overlapping decompositions. In order to improve the convergence and to be able to use non-overlapping decompositions, it has been proposed to use more general boundary conditions. It is even possible to optimize them with respect to the efficiency of the method. Theoretical and numerical results are given along with open problems.

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Domain Decomposition Methods for Nonlinear Problems

Xiao-Chuan Cai Abstract:

In this talk we review and discuss some overlapping domain decomposition methods for solving nonlinear systems of algebraic equations arising from the discretization of partial differential equations. These methods are well studied as linear preconditioners in the context of Newton-Krylov based nonlinear solvers, and the focus of the talk is on extensions of the methods for nonlinear problems, in particular, for improving the nonlinear convergence of Newton's methods. Several classes of equations will be discussed and the emphasis is on coupled systems of nonlinear partial differential equations with local high nonlinearities.

Space-Time parallel Methods

Laurence Halpern

Abstract:

Evolution problems have a particular direction, namely the time direction, which usually plays quite a different role from the spatial directions. This needs to be taken into account when one tries to solve such problems in parallel. Over the last decades, several different approaches for the parallelization of evolution problems have been proposed and analyzed: shooting methods, parallel predictor correctior methods, waveform relaxation methods, parallel time stepping methods, space-time multigrid methods, and very recently the parareal algorithm, which fits into the class of shooting methods.

After a historical overview of these approaches, we will focus on the class of optimized Schwarz waveform relaxation methods. These methods are based on a decomposition of the problem in space, like classical Schwarz methods, but they solve subdomain problems in both space and time. This approach allows us to use non-matching grids both in space and time, or even different space-time models in different subdomains. Rapid convergence is obtained using optimized transmission conditions between subdomains, like in optimized Schwarz methods. Such methods are also easy to use, if one has already a solver for the associated evolution problem.