RMMM 2019Reliable Methods of Mathematical Modeling

September 9–13, 2019 – TU Wienhttps://www.asc.tuwien.ac.at/rmmm2019/

The biennial RMMM conferences bring together scientists developing reliable methods for mathematicalmodeling. The topics of the conference include applied and numerical analysis, methods for the controlof modeling and numerical errors, algorithmic aspects, challenging applications, and novel discretizationmethods for the numerical approximation of PDEs.

Conference venue

• Conference hall: Kontaktraum

• Building: TU Wien, Neues EI, Stiege I, 6th floor

• Address: Gußhausstraße 27–29, 1040 Vienna

Scientific committee

• Carsten Carstensen, Humboldt-Universität zu Berlin, Germany

• Dirk Praetorius, TU Wien, Austria

• Sergey I. Repin, Steklov Mathematical Institute, Russia

• Stefan Sauter, University of Zurich, Switzerland

Local organizing committee

• Gregor Gantner, TU Wien, Austria

• Dirk Praetorius, TU Wien, Austria

• Michele Ruggeri, TU Wien, Austria

ACKNOWLEDGMENTS

The support of the Austrian Science Fund (FWF) through the Doctoral School (DK) Dissipation anddispersion in nonlinear PDEs (grant W1245) and the Special Research Program (SFB) Taming com-plexity in partial differential systems (grant F65), as well as of TU Wien, the Vienna Center for PartialDifferential Equations, and the Central European Network for Teaching and Research in Academic Liaison(CENTRAL) is thankfully acknowledged.

n Vienna

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CONFERENCE PROGRAM AT A GLANCE

Mon 9 Sep Tue 10 Sep Wed 11 Sep Thu 12 Sep Fri 13 Sep09:00–09:30 Perugia Stevenson Bespalov Stephan09:30–10:00 Dong Karkulik Innerberger Gutleb10:00–10:30 Visinoni Zank Aretaki Jetta10:30–11:15 Coffee Coffee Coffee Coffee11:15–11:45 Harbrecht Picasso Repin Kreuzer11:45–12:15 Pfeiler Banas Sebastian Storn12:15–12:45 Mulita Rieder Bohn Bernkopf

12:45–14:30Registration(from 13:30)& Opening

Lunch break(individual)

Lunch break(individual)

Lunch break(individual)

Closing &Coffee

14:30–15:00 Bartels Carstensen Führer Heuer15:00–15:30 Egger Gedicke Van Venetië Erath15:30–16:00 Heid He Stocek Yu16:00–16:45 Coffee & Cake Coffee & Cake Coffee & Cake Coffee & Cake16:45–17:15 Vorhalík Malkovich Schöberl Toulopoulos17:15–17:45 Geevers Cranny Zhang Vasiliyev

17:45–18:15 Endtmayer Hang KalluciYousefikhosh-

bakhtGet together(from 18:15)

Social dinner(from 19:30)

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DETAILED CONFERENCE PROGRAM

Monday, September 9, 201913:30–14:30 Registration & Opening

14:30–15:00Sören Bartels (University of Freiburg, Germany)Simulation of twisted rods (page 9)

15:00–15:30Herbert Egger (TU Darmstadt, Germany)Systematic discretization of some nonlinear evolution problems (page 10)

15:30–16:00Pascal Heid (University of Bern, Switzerland)Adaptive iterative linearization Galerkin methods for nonlinear PDE (page 11)

16:00–16:45 Coffee & Cake

16:45–17:15Martin Vohralík (Inria Paris, France)A posteriori error estimates and adaptivity taking into account algebraic errors (page 12)

17:15–17:45Sjoerd Geevers (University of Vienna, Austria)An a posteriori error estimator for arbitrary-order Nédélec elements (page 13)

17:45–18:15Bernhard Endtmayer (RICAM Austrian Academy of Sciences, Austria)Efficiency and reliability for the DWR method (page 14)

from 18:15 Get together

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Tuesday, September 10, 2019

09:00–09:30Ilaria Perugia (University of Vienna, Austria)Adaptive hp discontinuous Galerkin methods for the Helmholtz equation (page 15)

09:30–10:00Zhaonan Dong (FORTH, Greece)A posteriori error estimates for hp-version discontinuous Galerkin methods on polygonaland polyhedral meshes (page 16)

10:00–10:30Michele Visinoni (University of Milano-Bicocca, Italy)A virtual element method for 3D elasticity problem based on the Hellinger–Reissnerprinciple (page 17)

10:30–11:15 Coffee

11:15–11:45Helmut Harbrecht (University of Basel, Switzerland)Modelling and simulation of partial differential equations on random domains (page 18)

12:15–12:45Carl-Martin Pfeiler (TU Wien, Austria)Dörfler marking with minimal cardinality is a linear complexity problem (page 19)

11:45–12:15Ornela Mulita (SISSA, Italy)Smoothed adaptive finite element method (page 20)

12:45–14:30 Lunch break (individual)

14:30–15:00Carsten Carstensen (Humboldt-Universität zu Berlin, Germany)Skeletal schemes for eigenvalue localisation? (page 21)

15:00–15:30Joscha Gedicke (University of Vienna, Austria)Benchmark computation of eigenvalues with large defect for non-selfadjoint ellipticdifferential operators (page 22)

15:30–16:00Xing-Shi He (Xi’an Polytechnic University, China)Parameter estimation by forward modelling and optimization (page 23)

16:00–16:45 Coffee & Cake

16:45–17:15Evgeny Malkovich (Sobolev Institute of Mathematics, Russia)Particles sintering in dense sphere packings (page 24)

17:15–17:45Ronan Cranny (ONERA Toulouse, France)Accurate method for calculating currents in wires in the vicinity of curved geometries(page 25)

17:45–18:15Xudeng Hang (Institute of Applied Physics and Computational Mathematics, China)A spherically-symmetric Lagrangian scheme for three dimensional hydrodynamics(page 26)

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Wednesday, September 11, 2019

09:00–09:30Rob Stevenson (University of Amsterdam, Netherlands)Stability of Galerkin discretizations of mixed space-time variational formulations ofparabolic evolution equations (page 27)

09:30–10:00Michael Karkulik (Universidad Técnica Federico Santa María, Chile)Space-time least squares finite elements for parabolic equations and applications(page 28)

10:00–10:30Marco Zank (University of Vienna, Austria)Realisation of a space-time continuous Galerkin finite element method for the heatequation in anisotropic Sobolev spaces (page 29)

10:30–11:15 Coffee

11:15–11:45Marco Picasso (EPFL, Switzerland)An adaptive space-time algorithm for a two-phase incompressible immiscible flow(page 30)

11:45–12:15Lubomir Banas (Bielefeld University, Germany)Robust space-time a posteriori estimates for the non-smooth Cahn–Hilliard equation(page 31)

12:15–12:45Alexander Rieder (TU Wien, Austria)hp-FEM for fractional parabolic problems (page 32)

12:45–14:30 Lunch break (individual)

14:30–15:00Thomas Führer (Pontificia Universidad Católica de Chile, Chile)Optimal quasi-diagonal preconditioners for pseudodifferential operators of order minustwo (page 33)

15:00–15:30Raymond van Venetië(University of Amsterdam, Netherlands)Optimal Calderón preconditioning without a dual mesh construction (page 34)

15:30–16:00Jakub Stocek (Heriot–Watt University, UK)Optimal operator preconditioning for pseudodifferential boundary problems on adaptivemeshes (page 35)

16:00–16:45 Coffee & Cake

16:45–17:15Joachim Schöberl (TU Wien, Austria)Pressure robust a posteriori error estimates for pressure robust methods (page 36)

17:15–17:45Shuo Zhang (Chinese Academy of Sciences, China)Some nonstandard high-efficiency finite element schemes (page 37)

17:45–18:15Eglantina Kalluci (University of Tirana, Albania)Parallel implementation of finite element methods (page 38)

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Thursday, September 12, 2019

09:00–09:30Alex Bespalov (University of Birmingham, UK)Convergence analysis of adaptive stochastic Galerkin FEM (page 39)

09:30–10:00Michael Innerberger (TU Wien, Austria)Instance-optimal goal-oriented adaptivity (page 40)

10:00–10:30Aikaterini Aretaki (National Technical University of Athens, Greece)Preconditioning techniques applied on optimal control problems and discretized by afictitious domain FEM with cut elements (page 41)

10:30–11:15 Coffee

11:15–11:45Sergey Repin (Steklov Mathematical Institute, Russia)Error analysis of free boundary problems (page 42)

11:45–12:15Daniel Sebastian (TU Wien, Austria)Functional a posteriori error estimates for boundary element methods (page 43)

12:15–12:45Jan Bohn (Karlsruhe Institute of Technology, Germany)A convergent finite element boundary element scheme for Maxwell–Landau–Lifshitz–Gilbert equations (page 44)

12:45–14:30 Lunch break (individual)

14:30–15:00Norbert Heuer (Pontifical Catholic University of Chile, Chile)DPG theory and techniques for plate bending problems (page 45)

15:00–15:30Christoph Erath (TU Darmstadt, Germany)On the nonsymmetric coupling method for parabolic-elliptic interface problems(page 46)

15:30–16:00Yunlong Yu (Institute of Applied Physics and Computational Mathematics, China)A finite volume scheme preserving maximum principle for the system of radiation diffu-sion equations with three-temperature (page 47)

16:00–16:45 Coffee & Cake

16:45–17:15Ioannis Toulopoulos (AC2T research GmbH & JKU Linz, Austria)Numerical methods for viscoplastic models in metal forming processes (page 48)

17:15–17:45Sergey Vasilyev (RUDN University, Russia)Layer-adapted piecewise uniform Shishkin-type meshes for solving boundary problemsof the singular perturbated fourth-order differential equation (page 49)

17:45–18:15Majid Yousefikhoshbakht (Bu-Ali Sina University, Iran)A meta-heuristic crow search algorithm for solving the open vehicle routing problemwith time windows (page 50)

from 19:30 Social dinner (page 8)

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Friday, September 13, 2019

09:00–09:30Ernst P. Stephan (Leibniz University Hannover, Germany)hp-FEM for variational inequalities of the second kind involving the p-Laplacian(page 51)

09:30–10:00Timon S. Gutleb (Imperial College London, UK & University of Vienna, Austria)A sparse spectral method for Volterra integral equations using orthogonal polynomialson the triangle (page 52)

10:00–10:30Mahipal Jetta (Mahindra École Centrale, India)An efficient explicit scheme for a fourth order nonlinear diffusion filter (page 53)

10:30–11:15 Coffee

11:15–11:45Christian Kreuzer (TU Dortmund, Germany)Recovery of conformity: quasi-optimal and pressure robust discretisations of the Stokesequations (page 54)

11:45–12:15Johannes Storn (Bielefeld University, Germany)Computation of the LBB constant with a least-squares finite element method (page 55)

12:15–12:45Maximilian Bernkopf (TU Wien, Austria)Optimal convergence rates in L2 for a first order system least squares finite elementmethod (page 56)

12:45–14:30 Closing & Coffee

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CONFERENCE DINNER

The conference dinner will take place at the restaurant Silberwirt.

SilberwirtSchloßgasse 21, 1050 Viennahttps://www.silberwirt.at

We meet at 19:00 in front of the main entrance of the Neues EI to walk to the dinner location (see thepath below).

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ABSTRACTS(chronological order)

Simulation of twisted rodsSören Bartels1 and Philipp Reiter2

1Department of Applied Mathematics, University of Freiburg, Germany2 Institute for Mathematics, Martin Luther University Halle-Wittenberg, Germany

A physical wire can be modeled by a framed curve. We assume that its behavior is driven by acombination of bending energy and twist energy. The latter tracks the rotation of the frame about thecenterline of the curve. To obtain a more realistic setting, we have to preclude self-intersections of thecurve which can be achieved by adding a self-avoiding term. We discuss the discretization of this modeland present numerical simulations. The work extends previous results from [1].

References

[1] S. Bartels, P. Reiter, Stability of a simple scheme for the approximation of elastic knots and self-avoiding inextensible curves, arXiv:1804.02206, 2018.

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Systematic discretization of some nonlinear evolution problemsHerbert Egger1

1Department of Mathematics, TU Darmstadt, Germany

We consider the systematic discretization of certain classes of nonlinear evolution problems whichare governed by energy or entropy dissipation. We will demonstrate that, after rewriting the problemson the continuous level in a particular form, the underlying energy or entropy dissipation structure canbe preserved automatically by Galerkin approximation in space and discontinuous or Petrov Galerkinmethods in time, yielding energy or entropy stable methods of formally arbitrary approximation order.The applicability of our approach will be illustrated by a brief discussion of several test problems, includingcross-diffusion systems, nonlinear electromagnetics, and rather general dissipative Hamiltonian systems,and some relations to other discretization methods will be highlighted.

References

[1] H. Egger, Structure preserving approximation of dissipative evolution problems, Numer. Math., 2019.DOI:10.1007/s00211-019-01050-w

[2] H. Egger, Energy stable Galerkin approximation of Hamiltonian and gradient systems, arX-ive:1812.04253, 2018.

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Adaptive iterative linearization Galerkin methods for nonlinear PDEPascal Heid1 and Thomas P. Wihler1

1Mathematics Institute, University of Bern, Switzerland

A wide variety of (fixed-point) iterative methods for the solution of nonlinear equations (in Hilbertspaces) exists. In many cases, such schemes can be interpreted as iterative local linearization methods,which can be obtained by applying a suitable linear preconditioning operator to the original (nonlinear)equation. Based on this observation, we will derive a unified abstract framework which recovers someprominent iterative schemes. Furthermore, in the context of numerical solutions methods for nonlinearpartial differential equations, we propose a combination of the iterative linearization approach and theclassical Galerkin discretization method, thereby giving rise to the so-called iterative linearization Galerkin(ILG) methodology. Moreover, still on an abstract level, based on elliptic reconstruction techniques, wederive a posteriori error estimates which separately take into account the discretization and linearizationerrors. Subsequently, we propose an adaptive algorithm, which provides an efficient interplay betweenthese two effects.

References

[1] P. Heid, T. P. Wihler, Adaptive Iterative Linearization Galerkin Methods for Nonlinear Problems,arXiv:1808.04990, 2018.

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A posteriori error estimates and adaptivity taking into accountalgebraic errorsMartin Vohralík1,2

1SERENA, Inria Paris, France2CERMICS, Université Paris-Est (ENPC), France

This talk addresses the derivation of a posteriori error estimates and of adaptive strategies for numericaldiscretizations of partial differential equations when the solution of the underlying (large sparse) systems oflinear algebraic equations is taken into account. I will start with the model Laplace equation and show howguaranteed upper and lower bounds on the total error can be obtained, while also developing guaranteedupper and lower bounds on both the discretization and algebraic error components, following [4, 5, 6].These results lead to safe stopping criteria for algebraic solvers that guarantee that the algebraic error doesnot dominate the total error, while avoiding unnecessary iterations. An hp-refinement strategy in presenceof inexact solvers is then described, following [3]. It specifically yields a computable guaranteed bound onthe error reduction factor between the consecutive hp-refinement steps. A generic framework for arbitrary(residual) functionals is then presented following [1]. Concrete applications are considered finally, namelyadaptive inexact iterative algorithms for the Stokes problem following [2], and an industrial application tomulti-phase multi-compositional porous media Darcy flows following [7]. In the latter framework, I alsodiscuss some implementation aspects covering arbitrary polytopal grids.

References

[1] J. Blechta, J. Málek, and M. Vohralík, Localization of the W−1,q norm for local a posteriori efficiency,IMA J. Numer. Anal., 2019. DOI:10.1093/imanum/drz002

[2] M. Čermák, F. Hecht, Z. Tang, and M. Vohralík, Adaptive inexact iterative algorithms based onpolynomial-degree-robust a posteriori estimates for the Stokes problem, Numer. Math. 138(4):1027–1065, 2018. DOI:10.1007/s00211-017-0925-3

[3] P. Daniel, A. Ern, and M. Vohralík, An adaptive hp-refinement strategy with inexact solvers andcomputable guaranteed bound on the error reduction factor, HAL Preprint 01931448, 2018.

[4] A. Miraçi, J. Papež, and M. Vohralík, A multilevel algebraic error estimator and the correspondingiterative solver with p-robust behavior, HAL Preprint 02070981, 2019.

[5] J. Papež, U. Rüde, M. Vohralík, and B. Wohlmuth, Sharp algebraic and total a posteriori errorbounds for h and p finite elements via a multilevel approach, HAL Preprint 01662944, 2017.

[6] J. Papež, Z. Strakoš, and M. Vohralík, Estimating and localizing the algebraic and total numericalerrors using flux reconstructions, Numer. Math. 138(3):681–721, 2019. DOI:10.1007/s00211-017-0915-5

[7] M. Vohralík, S. Yousef, A simple a posteriori estimate on general polytopal meshes with applica-tions to complex porous media flows, Comput. Methods Appl. Mech. Engrg. 331:728–760, 2018.DOI:10.1016/j.cma.2017.11.027

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An a posteriori error estimator for arbitrary-order Nédélec elementsJoscha Gedicke1, Sjoerd Geevers1, and Ilaria Perugia1

1Department of Mathematics, University of Vienna, Austria

We present a reliable a posteriori error estimator for Nédélec elements for the curl-curl problem thatdoes not involve a generic constant. The error estimator is based on equilibration of the curl field bysolving very local problems, resulting in very cheap computations that can be done in parallel. Such atype of error estimator was already introduced by Braess and Schöberl in [1] for the lowest-order Nédélecelement and requires solving local problems on small patches of elements known as vertex patches. Thenovelty of our estimator is that it can be applied to Nédélec elements of arbitrary degree. Furthermore,our estimator does not require solving problems on vertex patches, but instead requires solving problemson only single elements, single faces, and very small sets of nodes. Numerical examples also confirm thereliability of the estimator and show that its efficiency is very high, with an overshoot of no more than afactor 2 in all test cases.

References

[1] D. Braess and J. Schöberl, Equilibrated residual error estimator for edge elements, Math. Comp.77(262):651–672, 2008. DOI:10.1090/S0025-5718-07-02080-7

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Efficiency and reliability for the DWR methodBernhard Endtmayer1, Ulrich Langer1, and Thomas Wick2,3

1RICAM, Austrian Academy of Sciences, Austria2 Institute of Applied Mathematics, Leibniz Universität Hannover, Germany

3 Cluster of Excellence PhoenixD (Photonics, Optics, and Engineering - Innovation Across Disciplines), LeibnizUniversität Hannover, Germany

In this talk, we derive two-sided a posteriori error estimates for the dual-weightedresidual (DWR)method. We consider single and multiple quantites of interest for estimation. Using a saturation assump-tion for the quantity of interest, we state lower bounds and upper bounds for the error estimator yieldingefficiency and reliability. This is done for large class of nonlinear Problems and nonlinear quantities ofinterest. We also perform careful studies of the remainder term that is usually neglected. Finally weprovide some numerical examples.

This work has been supported by the Austrian Science Fund (FWF) under the grant P 29181 Goal-Oriented Error Control for Phase-Field Fracture Coupled to Multiphysics Problems.

References

[1] B. Endtmayer, U. Langer, and T. Wick, Two-side a posteriori error estimates for the DWR method,arXiv:1811.07586, 2018.

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Adaptive hp discontinuous Galerkin methods for the Helmholtzequation

Scott Congreve1, Joscha Gedicke2, and Ilaria Perugia2

1Faculty of Mathematics and Physics, Charles University Prague, Czech Republic2Faculty of Mathematics, University of Vienna, Austria

The hp adaptive refinement procedure for discontinuous Galerkin (DG) discretizations of the Helmholtzequation introduced and analyzed in [1] will be presented.

The Helmholtz equation with impedance boundary condition in two space dimensions will be taken asa model problem, and polynomial-based DG approximations with stabilization terms containing jumps ofthe traces, as well as jumps of the normal components of the gradients at the interelement boundaries,will be considered. These methods are unconditionally well-posed [2], and are therefore well-suited alsowhen the adaptive procedure is started in the pre-asymptotic regime.

The a posteriori error analysis we will present is based on a local reconstruction of equilibratedfluxes [3, 4] applied to an auxiliary shifted Poisson problem with inhomogeneous Neumann boundaryconditions. With respect to the standard framework of equilibrated fluxes for elliptic problems, additionalterms related to the DG treatment of inhomogeneous impedance boundary conditions appear in theestimator, together with an extra lifting operator, which is required in the definition of the discretegradient. The presence of this term is due to the additional gradient stabilization terms in the DGdiscretisation of the Helmholtz problem. The error due to the nonconformity of the approximation spacesis controlled by a potential reconstruction.

The considered a posteriori error estimator is proven to be both reliable and efficient, outside theregime of pollution, up to generic constants which are independent of the wave number, the polynomialdegrees, and the element sizes. By construction, the estimator captures possible singularities of thesolution correctly. With an arguement from [5], it is proven that, in contrast to residual based a posteriorierror estimators, the presented error estimator is robust in the polynomial degree. Some numericalexperiments will be presented, in order to confirm the efficiency of hp-adaptive refinement strategiesbased on the presented a posteriori error estimator.

References

[1] S. Congreve, J. Gedicke, and I. Perugia, Robust adaptive hp discontinuous Galerkin finite el-ement methods for the Helmholtz equation, SIAM J. Sci. Comput. 41(2):A1121–A1147, 2019.DOI:10.1137/18M1207909

[2] M. Melenk, A. Parsania, and S. Sauter, General DG-methods for highly indefinite Helmholtz prob-lems, J. Sci. Comput. 57(3):536–581, 2013. DOI:10.1007/s10915-013-9726-8

[3] V. Dolejší, A. Ern, and M. Vohralík, hp-adaptation driven by polynomial-degree-robust a pos-teriori error estimates for elliptic problems, SIAM J. Sci. Comput. 38(5):A3220–A3246, 2016.DOI:10.1137/15M1026687

[4] A. Ern and M. Vohralík, Polynomial-degree-robust a posteriori estimates in a unified setting forconforming, nonconforming, discontinuous Galerkin, and mixed discretizations, SIAM J. Numer.Anal. 53(2):1058–1081, 2015. DOI:10.1137/130950100

[5] D. Braess, V. Pillwein, and J. Schöberl, Equilibrated residual error estimates are p-robust, Comput.Methods Appl. Mech. Engrg. 198(13-14):1189–1197, 2009. DOI:10.1016/j.cma.2008.12.010

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A posteriori error estimates for hp-version discontinuous Galerkinmethods on polygonal and polyhedral meshes

Andrea Cangiani1, Zhaonan Dong2, and Emmanuil H. Georgoulis3,4

1School of Mathematical Sciences, University of Nottingham, UK2Institute of Applied and Computational Mathematics, FORTH, Greece

3 Department of Mathematics, University of Leicester, UK4 Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical

University of Athens, Greece

We present a new a posteriori error analysis for hp-version interior penalty discontinuous Galerkin (dG)methods for linear elliptic problems. The a posteriori error bounds are proven for meshes consisting ofextremely general polygonal/polyhedral element shapes. In particular, arbitrary number of very small facesare allowed on each polygonal/polyhedral element, as long as certain mild shape regularity assumptionsare satisfied. The case of simplicial and/or box-type elements is included in the analysis as a special case.As such the present analysis generalizes the know a posteriori error analysis results for hp-dG methodsto admit arbitrary number of irregular hanging nodes per element. The proof hinges on a new recoverystrategy in conjunction with a generalized Helmholtz decomposition formula. The resulting a posteriorierror bound involves jumps on the tangential derivatives along elemental faces. Local lower bounds arealso proven, indicating the optimality of the proposed approach. A series of numerical experiments is alsopresented, highlighting the good performance of the a posteriori error bounds.

References

[1] A. Cangiani, Z. Dong, and E. Georgoulis, A posteriori error estimates for hp-version discontinuousGalerkin methods on polygonal and polyhedral meshes, in preparation.

[2] A. Cangiani, Z. Dong, E. Georgoulis, and P. Houston, hp-Version Discontinuous Galerkin Methodson Polygonal and Polyhedral Meshes, SpringerBriefs Math., 2017. DOI:10.1007/978-3-319-67673-9

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A virtual element method for 3D elasticity problem based on theHellinger-Reissner principle

Franco Dassi1, Carlo Lovadina2,3, and Michele Visinoni1

1Dipartimento di Matematica e Applicazioni, Università degli studi di Milano Bicocca, Italy2Dipartimento di Matematica, Università degli studi di Milano, Italy

3IMATI del CNR, Via Ferrata 1, 27100 Pavia, Italy

The elasticity theory deals with deformation of bodies under the influence of applied forces, andin particular, with stresses and strains which result from deformations. Within the framework of smalldisplacements and small deformations, we take the Hellinger-Reissner variational principle as the basisof the discretization procedure. This mixed formulation, based on the Hellinger-Reissner functional,describes the problem by catching the displacements and the stresses as variables. It is well-known that inthe Finite Element practice, designing a stable and accurate scheme, which preserves both the symmetryof the stress tensor and the continuity of the tractions at the inter-element, is commonly recognized asa difficult task. The fundamental reason behind this difficulty lies in the local polynomial approximation.Therefore, we exploit the flexibility of Virtual Element Methods to avoid these troubles and to developa valid alternative which is reasonably cheap with respect to deserved accuracy. Recently, some VirtualElement schemes have been proposed and analyzed for two-dimensional problems [1, 2]. In this talk, wepresent an extension to the three-dimensional case for the low-order VEM scheme in [1]. The aim is topropose a valid alternative to the corresponding mixed Finite Element with H(div)-conformity for stressand L2-conformity for displacements. Some numerical tests are provided in order to show the validity andthe potential of our analysis.

References

[1] E. Artioli, S. de Miranda, C. Lovadina, L. Patruno, A stress/displacement Virtual Elementmethod for plane elasticity problems Comput. Methods Appl. Mech. Engrg. 325:155–174, 2017.DOI:10.1016/j.cma.2017.06.036

[2] E. Artioli, S. de Miranda, C. Lovadina, L. Patruno, A family of virtual element methods for planeelasticity problems based on the Hellinger-Reissner principle, Comput. Methods Appl. Mech. Engrg.340:978-999, 2018. DOI:10.1016/j.cma.2018.06.020

[3] F. Dassi, C. Lovadina, M. Visinoni, A three-dimensional Hellinger-Reissner Virtual Element Methodfor linear elasticity problems, arXiv:1906.06119, 2019.

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Modelling and simulation of partial differential equations on randomdomains

Helmut Harbrecht1

1Department of Mathematics and Computer Science, University of Basel, Switzerland

There are basically two methods to deal with partial differential equations on random domains. On theone hand, the perturbation method is based on a prescribed perturbation field of the boundary and usesa shape Taylor expansion with respect to this perturbation field to approximately represent the randomsolution. This yields a simple approach which, however, induces a model error. On the other hand, inthe domain mapping approach, the random domain is mapped on a nominal, fixed domain. This requiresthat the perturbation field is also known in the interior of the domain but the resulting partial differentialequation with random diffusion matrix and random load can be solved without systematic errors. Inthis talk, we present theoretical and practical results for both methods. In particular, we discuss theiradvantages and disadvantages.

References

[1] M. Dambrine, I. Greff, H. Harbrecht, and B. Puig, Numerical solution of the Poisson equa-tion with a thin layer of random thickness, SIAM J. Numer. Anal., 54(2):921–941, 2016.DOI:10.1137/140998652

[2] H. Harbrecht, N. Ilić, and M.D. Multerer, Rapid computation of far-field statistics for random obstaclescattering, Eng. Anal. Bound. Elem. 101:243–251, 2019. DOI:10.1016/j.enganabound.2018.11.005

[3] H. Harbrecht, M. Peters, and M. Siebenmorgen. Analysis of the domain mapping method for ellipticdiffusion problems on random domains, Numer. Math. 134(4):823–856, 2016. DOI:10.1007/s00211-016-0791-4

[4] H. Harbrecht and M. Schmidlin. Multilevel quadrature for elliptic problems on random domains bythe coupling of FEM and BEM. arXiv:1802.05966, 2018.

[5] H. Harbrecht, R. Schneider, and C. Schwab, Sparse second moment analysis for elliptic problems instochastic domains, Numer. Math. 109(3):385–414, 2008. DOI:10.1007/s00211-008-0147-9

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Dörfler marking with minimal cardinalityis a linear complexity problemCarl-Martin Pfeiler1 and Dirk Praetorius1

1Institute for Analysis and Scientific Computing, TU Wien, Austria

Adaptive finite element methods (AFEM) iterate the procedure Solve-Estimate-Mark-Refine in orderto generate a sequence of locally refined meshes (T`)`∈N0 , where the degrees of freedom are chosen morecarefully than for uniform mesh refinement: First, the discrete solution is computed on the given mesh T`.Then, local refinement indicators η`(T ) are computed for all T ∈ T`. Based on these, a subsetM` ⊆ T`is marked for refinement. Finally, (at least) the marked elements are refined to obtain an improved meshT`+1.

In his seminal work [1], Dörfler proposes a marking criterion, which allows to prove linear convergenceof the energy error for each iteration of the AFEM algorithm: Given a marking parameter 0 < θ ≤ 1,construct a setM` ⊆ T` such that

θ∑

T∈T`

η`(T )2 ≤∑

T∈M`

η`(T )2 .

Later it was shown by Stevenson [2] that the Dörfler marking criterion is not only sufficient to prove linearconvergence, but even in some sense necessary.

In the literature, different algorithms have been proposed to constructM`, where usually two goalscompete: On the one hand,M` should contain a minimal number of elements. On the other hand, oneaims for linear costs O(#T`) with respect to the number of elements. Unlike expected in the literature [2],we formulate and analyze an algorithm, which constructs a minimal setM` at linear costs. In particular,Dörfler marking with minimal cardinality is a linear complexity problem.

References

[1] W. Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal.33(3):1106–1124, 1996. DOI:10.1137/0733054

[2] R. Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math.7(2):245–269, 2007. DOI:10.1007/s10208-005-0183-0

19

Smoothed adaptive finite element methodOrnela Mulita1, Stefano Giani2, and Luca Heltai1

1mathLab Laboratory, Scuola Internazionale Superiore di Studi Avanzati, Italy2Department of Engineering, Durham University, UK

We propose a new algorithm for Adaptive Finite Element Methods based on Smoothing iterations(S-AFEM). The algorithm is inspired by the ascending phase of the V-cycle multigrid method: we re-place accurate algebraic solutions in intermediate cycles of the classical AFEM with the application of aprolongation step, followed by a fixed number of few smoothing steps. The method reduces the overallcomputational cost of AFEM by providing a fast procedure for the construction of a quasi-optimal meshsequence with large algebraic error in the intermediate cycles. Indeed, even though the intermediate solu-tions are far from the exact algebraic solutions, we show that their a-posteriori error estimation produces arefinement pattern that is substantially equivalent to the one that would be generated by classical AFEM,at a considerable fraction of the computational cost. In this talk, we will quantify rigorously how the errorpropagates throughout the algorithm, and then we will provide a connection with classical a posteriorierror analysis. Finally, we will present a series of numerical experiments that highlights the efficiency andthe computational speedup of S-AFEM.

References

[1] O. Mulita, S. Giani, and L. Heltai, Quasi-optimal mesh sequence construction through SmoothedAdaptive Finite Element Method, arXiv:1905.06924, 2019.

20

Skeletal schemes for eigenvalue localisation?Carsten Carstensen1

1Department of Mathematics, Humboldt-Universität zu Berlin, Germany

The localisation of eigenvalue is one of the fundamental tasks in numerical analysis even for a Laplaceoperator and Dirichlet boundary conditions on a polyhedral bounded Lipschitz domain. The Rayleigh-Ritz principles for those symmetric PDE eigenvalue problems lead to (guaranteed) upper eigenvaluebounds with conforming finite elements even in view of inexact solve. A simple post-processing ofnonconforming schemes like the Crouzeix-Raviart (for the Laplace) or Morley (for the bi-Laplace) finiteelement methods leads to guaranteed lower eigenvalue bounds. The improved version of this result from[1, 2] will be presented and an attempt is made to generalise it to skeletal methods. This paper suggeststhe eigenvalue computation with a scheme that, for the source problem, belongs to the class of hybridizablediscontinuous Galerkin schemes with Lehrenfeld-Schöberl stabilisation and is also called a weak Galerkinscheme in the literature. The presentation discusses the appropriate choice of a stabilisation parameterand the computation of guaranteed eigenvalue bounds. The presentation also discusses new proofs of theasymptotic lower bound properties in [3].

The topics reflect joint work with Joscha Gedicke (U Vienna), Dietmar Gallistl (U Jena), with QilongZhai and Ran Zhang (Jilin U, China) as well as Sophie Puttkammer (Humboldt).

References

[1] C. Carstensen and J. Gedicke, Guaranteed lower bounds for eigenvalues, Math. Comp. 83:2605–2629,2014. DOI:10.1090/S0025-5718-2014-02833-0

[2] C. Carstensen and D. Gallistl, Guaranteed lower eigenvalue bounds for the biharmonic equation,Numer. Math. 126(1):33–51, 2014. DOI:10.1007/s00211-013-0559-z

[3] Q. Zhai and R. Zhang, Lower and upper bounds of Laplacian eigenvalue problem by weak Galerkinmethod on triangular meshes, Discrete Contin. Dyn. Syst. Ser. B 24:403–413, 2019.

21

Benchmark computation of eigenvalues with large defect fornon-selfadjoint elliptic differential operators

Rebekka Gasser1, Joscha Gedicke2, and Stefan Sauter1

1Institut für Mathematik, Universität Zürich, Switzerland2Fakultät für Mathematik, Universität Wien, Austria

In this talk we present benchmark problems for non-selfadjoint elliptic eigenvalue problems with largedefect and ascent. We describe the derivation of the benchmark problem with a discontinuous coefficientand mixed boundary conditions. Numerical experiments are performed to investigate the convergence of aGalerkin finite element method with respect to the discretization parameters, the regularity of the problem,and the ascent of the eigenvalue. This allows us to verify the sharpness of the theoretical estimates fromthe literature with respect to these parameters. We provide numerical evidence about the size of theascent and show that it is important to consider the mean value for the eigenvalue approximation.

References

[1] R. Gasser, J. Gedicke, and S. Sauter, Benchmark computation of eigenvalues with large defect fornon-selfadjoint elliptic differential operators, arXiv:1902.02114, 2019.

22

Parameter estimation by forward modelling and optimizationXing-Shi He1, Qin-Wei Fan1, and Xin-She Yang2

1College of Science, Xi’an Polytechnic University, China2School of Science and Technology, Middlesex university London, UK

Many problems in engineering and science involve parameter estimations of material properties andkey parameters for a system or underlying structures. For a given set of measurements of some quantities,such estimations are essentially inverse problems. Since data and measurements are usually incompletein practice, it is quite challenging to solve such inverse problems and solutions obtained may not beunique [1, 2, 3]. For a certain class of problems, it is possible to formulate the inverse parameterestimation as an optimization problem subject to various nonlinear constraints [2, 4].

The objective function can be the errors between the measured data D and the predicted responsesyi (x) that are calculated in an iterative, forward manner using a mathematical model in terms of a set ofdifferential equations and partial different equations. Constraints are imposed using limits of geometry,possible parameter ranges and other prior knowledge of the problem. Thus, the parameter estimationproblem becomes

Minimize f (p) =m∑

i=1

|yi (p)− Di |2, p = (p1, p2, ..., pn) ∈ <n, (1)

subject to constraints. Here, the unknown parameters form a vector p=(p1, p2, ..., pn). Then, we canin principle solve such optimization problems using optimization techniques [2, 3]. Though traditionalalgorithms may work in many cases, a major trend now is to use nature-inspired algorithms [4, 5].Traditional algorithms can be considered as local optimizers, and their final solutions may largely dependon the starting points. On the other hand, nature-inspired techniques tend to be global optimizers, andthey have a higher probability of finding the true global optimal solutions. In addition, these nature-inspired metaheuristic algorithms are usually flexible, easy to implement and yet sufficiently effective [5].

This paper uses a new population-based nature-inspired cuckoo search algorithm (CSA) to solve op-timization problems of parameter estimation with proper constraint handling techniques such as dynamicpenalty method. To test the proposed approach, we use three benchmark problems, including the param-eter estimation of a dynamic system with measured responses, a beam design problem, and an inverseproblem involving a partial differential equation. We show that nature-inspired cuckoo search can workbetter than traditional algorithms when the problem is highly nonlinear and the number of parametersis not too high. With some parametric studies and sensitivity analysis, we also show that this proposedapproach is robust and stable.

References

[1] R. Aster, B. Borchers, C. Thurber, Parameter Estimation and Inverse Problems, Third Edition,Elsevier, 2018. DOI:10.1016/C2009-0-61134-X

[2] S. Boyd, L. Vandenberge, Convex Optimization, Cambridge University Press, 2004.

[3] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The Art of ScientificComputing, Third Edition, Cambridge University Press, 2007.

[4] X.S. Yang, Nature-Inspired Optimization Algorithms, Elsevier, 2014. DOI:10.1016/C2013-0-01368-0

[5] X.S. Yang, X.S. He, Mathematical Foundation of Nature-Inspired Algorithms, Springer Briefs inOptimization, 2019. DOI:10.1007/978-3-030-16936-7

23

Particles sintering in dense sphere packingsYaroslav Bazaikin1,3, Vladimir Derevschikov2,3, and Evgeny Malkovich1,3

1Sobolev Institute of Mathematics, Russia2Boreskov Institute of Catalysis, Russia3Novosibirsk State University, Russia

Highly porous material of CaO sorbent can be described as a dense packing of spheres with radiiapproximately 100nm. During repetitive cycles of C O2 sorption/regeneration the body of the sorbentundergoes the sintering processes which lead to effectiveness decreasing. There are number of approachesto simulate the mass transfer during the sintering. For the first (up to 10th) cycles the most adequatemodel is surface mass transportation described by equation

∂φ

∂t= ν

dh

dt+

1

2ωh3∆H, (1)

where H is a mean curvature of the sintering CaO particles surface, h is a thickness of reacted C aCO3, νand ω are constants. Equation (1) can be derived using standard conservation laws [1]. For the surfaceof revolution which corresponds to the case of two tangent spheres equation (1) takes form

∂f

∂t= ν

dh

dt+ ωh3(t)

1

f

∂

∂s

(f∂H

∂s

). (2)

In (2) s is a natural parameter on the revolution curve. For the reaction between solid and gas phasesthe thickness of C aCO3 layer h(t) = 3

√t. Initial data f (x , 0) =

√2Rx − x2 is an circle arc of radius R .

The boundary conditions depends on the density of the packing via coordination number Z .

Evolution of sintering particles profile (left) and stages of particles sintering for coordination numberZ = 6 (right) [1].

Authors thank the Government of the Novosibirsk Region (Russian Federation) and RFBR for financial support(research project No. 19-43-543013).

References

[1] Y.V. Bazaikin, V.S. Derevschikov, E.G. Malkovich, A.I. Lysikov, A.G. Okunev, Evolution of sorptiveand textural properties of CaO-based sorbents during repetitive sorption/regeneration cycles: Part II.Modeling of sorbent sintering during initial cycles, Chemical Engineering Science 199:156-163, 2019.DOI:10.1016/j.ces.2018.12.065

24

Accurate method for calculating currents in wires in the vicinity ofcurved geometries

Ronan Cranny1, Xavier Ferrieres1, and Thibault Volpert1

1 ONERA/DEMR, Université de Toulouse, France

Precise methods to calculate currents are required for low frequency EMC simulations dealing with vehiclesstruck by lightning. The current model used resolves Maxwell’s equations combined with a Line model based onHolland’s thin wire formalism [1]. The challenge is related to the approximation of the source fields obtainedwith Yee’s scheme. These sources are then used for the thin wire equations. In the vicinity of structures, theerrors due to the staircase meshes representing surfaces corrupt the fields’ values. In order to bypass this issue, itwas suggested to apply non structured meshes such as Finite Volume (FV) [2]. Difficulties are encountered whenintroducing thin oblique wires [3] in this last approach, in particular for the calculation of the local self inductanceL, a numerical parameter required by the line model equations.In choosing a FV solver, difficulties will arise in terms of calculation resources due to the calculation procedureof the latter and to the unstructuredness of the meshes. To overcome this obstacle, a hybrid Non Structured-Structured (NST-ST) FV scheme which can also incorporate oblique Line models is proposed.To illustrate the advantage of this new approach, an open cylindrical structure with wires running along its wallswill be taken into account. It will be illuminated by a plane wave and we shall compare the obtained results interms of current and field values retrieved inside and also in the vicinity of the cables.

(a) FV scheme (b) Ey fields calculated with FD and FV schemes

Figure 1: Simulated system illuminated by a Gaussian plane wave

References

[1] R. Holland, L. Simpson, Finite-Difference Analysis of EMP Coupling to Thin Struts and Wires, IEEE Trans-actions on Electromagnetic Compatibility EMC-23(2), 1981. DOI: 10.1109/temc.1981.303899

[2] P. Bonnet, X. Ferrieres, F. Issac, F. Paladian, J. Grando, J.C. Alliot, J. Fontaine, Numerical Modeling ofScattering Problems Using a Time Domain Finite Volume Method, Journal of Electromagnetic Waves 11(8),1997. DOI:10.1163/156939397X01070

[3] C. Guiffaut, A. Reineix, B. Pecqueux, New Oblique Thin Wire Formalism in the FDTD Method With MultiwireJunctions, IEEE Trans. Antennas and Propagation 60(3), 2012. DOI:10.1109/tap.2011.2180304

25

A spherically-symmetric Lagrangian scheme for three dimensionalhydrodynamics

Xudeng Hang1,2, Shuai Wang1,2, Guangwei Yuan1, and Zhijun Shen1,2

1Institute of Applied Physics and Computational Mathematics, Beijing, China2Laboratory of Computational Physics, Beijing, China

It is important for Lagrangian schemes to preserve spherical symmetry. Many research works are devoted tothis topic. Most of them are related to the case of 2D r − z geometry, e.g. [3, 4]. Compared with 2D cases, thereare much less of works on the three dimensional Lagrangian schemes. The scheme of [1] can preserve sphericalsymmetry by modifying the direction of the force. There is no evidence that the scheme in [2] can preservespherical symmetry. To the best of our knowledge, there is no report of 3D spherical symmetric scheme withoutmodification of the force for Cartesian geometry.

In this paper, we propose a new staggered-mesh Lagrangian scheme for 3D hydrodynamics of Cartesiangeometry. The scheme is based on an novel discretizaion which adopts overlapping control volumes. For thespherically symmetric problems, the scheme can preserve the spherical symmetry exactly on the sperical mesh asshown in 2 without modifying the force direction. We will show the construction of the scheme and prove thespherical symmetry. Numerical experiments are presented to verify the properties of the scheme.

Figure 2: The meshes of the 3D Noh problem

References

[1] E.J. Caramana, C.L. Rousculp, D.E. Burto, A. Compatible, Energy and symmetry-preserving LagrangianHydrodynamics Algorithm in Three-Dimensional Cartesian Geometry, J. Comp. Phys., 157:89–119, 2000.DOI:10.1006/jcph.1999.6368

[2] R. Loubère, P.-H. Maire, P. Váchal, 3D staggered Lagrangian hydrodynamics scheme with cell-centeredRiemann solver-based artificial viscosity, Int. J. Numer. Meth. Fluids 72:22–42,2013. DOI:10.1002/fld.3730

[3] M. Kenamond, M. Bement, M. Shashkov, Compatible, total energy conserving and symmetry-preservingarbitrary Lagrangian–Eulerian hydrodynamics in 2D rz–Cylindrical coordinates, Journal of ComputationalPhysics, 268:154–185,2014. DOI:10.1016/j.jcp.2014.02.039

[4] L. Margolin, M. Shashkov, Using a Curvilinear Grid to Construct Symmetry-Preserving Discretizations forLagrangian Gas Dynamics, J. Comp. Phys., 149:389–417,1999. DOI:10.1006/jcph.1998.6161

26

Stability of Galerkin discretizations of mixed space-timevariational formulations of parabolic evolution equations

Rob Stevenson1 and Jan Westerdiep1

1 Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Netherlands

We present two well-posed mixed simultaneous space-time variational formulations of parabolic evolutionequations, and derive sufficient conditions for stability of their Galerkin discretizations. These conditions aresatisfied by finite element discretizations w.r.t. non-uniform meshes that allow a partitioning into time-slabs.For approximating singularities that are local in space and time, we present stable Galerkin discretizations fromsubspaces that are spanned by tensor products of wavelets-in-time, and finite elements in space.

References

[1] R. Stevenson, J. Westerdiep, Stability of Galerkin discretizations of a mixed space-time variational formulationof parabolic evolution equations, arXiv:1902.06279, 2019.

27

Space-time least squares finite elements for parabolic equations andapplications

Thomas Führer1 and Michael Karkulik2

1Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Chile2Departamento de Matemática, Universidad Técnica Federico Santa María, Chile

In the last couple of years, different space-time discretizations for parabolic problems have been proposed inthe literature. In our talk, based on our work [1], we present a space-time least squares finite element method forthe heat equation. The advantages of our method over current space-time approaches is that we do not make anyassumption on the space-time mesh (apart from the usual assumptions on spatial meshes), that our formulationis of Galerkin-type (which means that we do not have to worry about discrete inf-sup conditions), and that wehave an a-posteriori error estimator for free. In particular, our approach features full space-time adaptivity. Wewill present theory and numerical results.

References

[1] T. Führer, M. Karkulik. Space-time least squares finite elements for parabolic equations and applications, inpreparation, 2018.

28

Realisation of a space-time continuous Galerkin finite elementmethod for the heat equation in anisotropic Sobolev spaces

Olaf Steinbach1 and Marco Zank2

1Institut für Angewandte Mathematik, TU Graz, Austria2Faculty of Mathematics, University of Vienna, Austria

For the discretisation of time-dependent partial differential equations, the standard approaches are explicit orimplicit time stepping schemes together with finite element methods in space. An alternative approach is theusage of space-time methods, where the space-time domain is discretised and the resulting global linear system issolved at once. In this talk, the model problem is the heat equation. First, a space-time variational formulation inanisotropic Sobolev spaces for the heat equation is discussed, where a linear isometry HT is used such that ansatzand test spaces are equal. A conforming discretisation of this space-time variational formulation in anisotropicSobolev spaces leads to a Galerkin-Bubnov finite element method, which is unconditionally stable, i.e. no CFLcondition is required. However, for the implementation of this method, the realisation of the linear isometry HT

is crucial. The main part of this talk investigates possible ways of doing this realisation for piecewise polynomial,globally continuous ansatz and test functions. In the last part of the talk, numerical examples are shown anddiscussed.

References

[1] O. Steinbach and M. Zank, Coercive space-time finite element methods for initial boundary value problems,submitted, 2018.

[2] M. Zank, Inf-Sup Stable Space-Time Methods for Time-Dependent Partial Differential Equations, PhD thesis,TU Graz, 2019.

29

An adaptive space-time algorithmfor a two-phase incompressible immiscible flow

Samuel Dubuis1 and Marco Picasso1

1Institute of Mathematics, EPFL, Switzerland

An adaptive method is proposed for a two-phase incompressible immiscible flow. The time-dependent Navier-Stokes equations with variable density and viscosity are coupled to a transport equation for the phase. Continuous,piecewise linear finite elements with large aspect ratio are used for the space discretization, order two finitedifferences are used for the time discretization.

The error indicator is based on a posteriori error estimates derived for the transport [1] and the time-dependentNavier-Stokes equations.

Numerical results are presented, first on an academic test case, which allows the effectivity index to beinvestigated, then on more complex situations, such as Rayleigh-Taylor instabilities. In particular, we shall verifynumerically a theorem of Hillairet [2] which states: “We study the motion of a ball inside a cavity filled with anincompressible constant-density viscous fluid. Then, there is no collision between the ball and the boundary of thecavity”.

References

[1] S. Dubuis and M. Picasso, An Adaptive Algorithm for the Time Dependent Transport Equation withAnisotropic Finite Elements and the Crank–Nicolson Scheme, J. Sci. Comput. 75(1):350-375, 2018.DOI:10.1007/s10915-017-0537-1

[2] M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow, Comm. PartialDifferential Equations 32(9):1345-1371, 2007. DOI:10.1080/03605300601088740

30

Robust space-time a posteriori estimates for the non-smoothCahn–Hilliard equation

Ľubomír Baňas1 and Christian Vieth1

1Department of Mathematics, Bielefeld University, Germany

We consider the Cahn-Hilliard variational inequality

∂tu −∆w = 0 , (1)

w + ε∆u ∈ ε−1∂Ψ(u) ,

where u is an order parameter, w a chemical potential and ε is a small (interfacial width) parameter. The term∂Ψ is the subdifferential of a non-smooth potential

Ψ(u) :=

12 (1− u2) if u ∈ [−1, 1] ,∞ if u /∈ [−1, 1] .

(2)

Alternative (smooth) choices for Ψ , e.g., the double-well or the logarithmic potential, are available, cf. [4], [3] andthe references therein. Nevertheless, the use of the non-smooth potential (2) is attractive from practical point ofview due to the exact preservation of the physically meaningful constraint u ∈ [−1, 1] and the possibility to useefficient algebraic solvers, which in combination with adaptive mesh refinement effectively reduce the dimensionof the problem, cf. [1], [2].

Standard analysis of numerical approximations of the Cahn-Hilliard equation yields sub-optimal error estimatesthat depend exponentially on the inverse of the (small) parameter ε. Improved error estimate with polynomialdependence on ε−1 can be derived using lower bounds for the principal eigenvalue of the linearized Cahn-Hilliardequation, cf. [4]. The principal eigenvalue technique can also be used to derive a posteriori estimate for theCahn-Hilliard equation with smooth potentials, see [3] and the references therein.

Due to the lack of the principal eigenvalue property and the non-smoothness of the potential (2), the approachof [4], [3] is not directly transferable to (1). So far only a posteriori estimates for the the control of spatial errorof the time-discretized version of (1) have been available, cf. [2].

Using a smooth regularization procedure from [5] along with the ideas of [3] we derive computable and ε-robustspace-time a posteriori estimates for a fully discrete finite element approximation of the Cahn-Hilliard equationwith the regularized non-smooth potential (2). We show that under certain natural assumptions the estimatesalso remain valid for the non-smooth Cahn-Hilliard variational inequality (1). We present numerical experimentsto demonstrate the efficiency of the proposed approach.

References

[1] Ľ. Baňas, R. Nürnberg, Finite element approximation of a three dimensional phase field model for voidelectromigration, J. Sci. Comput. 37(2):202–232, 2008. DOI:10.1007/s10915-008-9203-y

[2] Ľ. Baňas, R. Nürnberg, A posteriori estimates for the Cahn-Hilliard equation with obstacle free energy, M2ANMath. Model. Numer. Anal. 43(5):1003–1026, 2009. DOI:10.1051/m2an/2009015

[3] S. Bartels, R. Müller, Error control for the approximation of Allen-Cahn and Cahn–Hilliard equations with alogarithmic potential, Numer. Math. 119(3):409–435, 2011. DOI:10.1007/s00211-011-0389-9

[4] X. Feng, A. Prohl, Numerical analysis of the Cahn-Hilliard equation and approximation of the Hele-Shawproblem, Interfaces Free Bound. 7(1):1–28, 2005. DOI:10.4171/IFB/111

[5] C. Kahle, An L∞ bound for the Cahn-Hilliard equation with relaxed non-smooth free energy, Int. J. Numer.Anal. Model., 14(2):243-254, 2017.

31

hp-FEM for fractional parabolic problemsJens Markus Melenk1 and Alexander Rieder1

1Institute for Analysis and Scientific Computing, TU Wien, Austria

In this talk, for parameters γ,β ∈ (0, 1], we consider time dependent fractional diffusion problems of the form

∂γt u +(−∆

)βu = f , u(0) = u0, u|∂Ω = 0.

If the data u0 and f do not satisfy the boundary conditions, singularities form at the boundary ∂Ω. We presenthow, by employing hp-Finite Element based discretizations, these difficulties can be resolved and one can obtainexponentially convergent numerical schemes.

Following [2], we present a numerical method based on a representation formula involving the Riesz-Dunfordfunctional calculus. We prove that, when combined with hp-FEM in space using techniques from [1], the resultingscheme delivers exponential convergence towards the exact solution without compatibility conditions on the data.

An alternative approach to apply hp-FEM techniques for such a non-local problem is to extend the spacialdimension by one using the so-called Cafarelli-Silvestre extension. This localizes (−∆)β leading to a degenerate,elliptic problem with dynamic boundary condition. This problem can then be treated using standard hp-FEM toolsas done [1].

References

[1] Jens Markus Melenk and Alexander Rieder, hp-FEM for the fractional heat equation, arXiv:1901.01767, 2019.

[2] A. Bonito, W. Lei, and J. E. Pasciak, Numerical Approximation of Space-Time Fractional Parabolic Equations,Comput. Methods Appl. Math. 17(4): 679–705, 2017. DOI:10.1515/cmam-2017-0032

32

Optimal quasi-diagonal preconditioners for pseudodifferentialoperators of order minus twoThomas Führer1 and Norbert Heuer1

1 Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Chile

In this talk I present results from our recent work [1]. We consider pseudodifferential operators of order minustwo in any space dimension and their discretization by piecewise polynomials.

We define and analyze quasi-diagonal preconditioners for the discretized operators where quasi-diagonal meansdiagonal up to sparse transformations. Considering shape regular simplicial meshes and arbitrary fixed polynomialdegrees, we prove, for dimensions larger than one, that our preconditioners are asymptotically optimal. Our analysisis based on the additive Schwarz theory and key ingredient is the decomposition of piecewise constants into thedivergence of lowest-order Raviart–Thomas functions.

Numerical experiments in two, three and four dimensions confirm our theoretical results. We present examplesfor uniform and adaptive mesh-refinement. Moreover, we show some applications of the preconditioner whichinvolve minimization problems in negative order Sobolev norms.

References

[1] T. Führer and N. Heuer, Optimal Quasi-diagonal Preconditioners for Pseudodifferential Operators of OrderMinus Two, J. Sci. Comput. 79(2):1161–1181, 2019. DOI:10.1007/s10915-018-0887-3

33

Optimal Calderón preconditioning without a dual mesh constructionRob Stevenson1 and Raymond van Venetië2

Korteweg–de Vries Institute for Mathematics, Faculty of Science, University of Amsterdam, Netherlands

We construct a preconditioner for negative and positive order operators, discretized by simplicial Lagrange ele-ments [4, 5]. The canonical cases are the Single Layer operator for piecewise constant functions, or the Hypersingu-lar operator for continuous piecewise linears. We propose a variation of the well-studied dual mesh preconditioningtechnique [1, 2, 3]. The resulting preconditioner yields a uniformly bounded condition number. Our approacheasily extends to operators discretized on locally refined partitions, and (dis)continuous elements of higher degree.

Compared to earlier proposals, the preconditioner has the following advantages: It does not require the inverseof a non-diagonal matrix; it applies without any mildly grading assumption on the partition; and it does not requirea barycentric refinement of the partition underlying the trial space.

The preconditioning strategy requires the application of an opposite order operator, e.g. for preconditioningof the Single Layer operator one can use the Hypersingular operator and vice versa. The total cost of thepreconditioner is the sum of the cost of the opposite order operator, and additional cost that is proportional tothe number of simplices.

We will conclude with numerical results for the Single Layer and Hypersingular operators.

References

[1] O. Steinbach and W. L. Wendland, The construction of some efficient preconditioners in the boundary elementmethod, Adv. Comput. Math. 9(1-2):191–216, 1998. DOI:10.1023/A:1018937506719

[2] R. Hiptmair, Operator preconditioning, Comput. Math. Appl. 52(5):699–706, 2006.DOI:10.1016/j.camwa.2006.10.008

[3] A. Buffa and S. Christiansen, A dual finite element complex on the barycentric refinement, Math. Comp.76(260):1743–1769, 2007. DOI:10.1090/S0025-5718-07-01965-5

[4] R. Stevenson and R. van Venetië, Optimal preconditioning for problems of negative order, accepted forpublication in Math. Comp., 2018. arXiv:1803.05226

[5] R. Stevenson and R. van Venetië, Optimal preconditioning for problems of positive order, arXiv:1906.09164,2019.

34

Optimal operator preconditioning for pseudodifferential boundaryproblems on adaptive meshes

Heiko Gimperlein1, Jakub Stocek1, and Carolina Urzúa Torres2

1Maxwell Institute, Heriot–Watt University, UK2Mathematical Institute, University of Oxford, UK

We propose an operator preconditioner for general elliptic pseudodifferential equations in a domain Ω, whereΩ is either in Rn or in a Riemannian manifold. For linear systems of equations arising from low-order Galerkindiscretizations, we obtain condition numbers that are independent of the mesh size and of the choice of bases fortest and trial functions.The basic ingredient is a classical formula by Boggio for the fractional Laplacian, which is extended analytically.In the special case of the weakly and hypersingular operators on a line segment or a screen, our approach gives aunified, independent proof for a series of recent results by Hiptmair, Jerez-Hanckes, Nédélec and Urzúa-Torres.We study the increasing relevance of the regularity assumptions on the mesh with the operator order. We disscussthe impact of these assumptions on adaptively generated meshes. Numerical examples validate our theoreticalfindings and illustrate the performance of the proposed preconditioner on quasi-uniform, graded and adaptivelygenerated meshes.

35

Pressure robust a posteriori error estimates for pressure robustmethods

Philip Lederer1, Christian Merdon2, and Joachim Schöberl1

1Institute for Analysis and Scientific Computing, TU Wien, Austria2WIAS, Berlin, Germany

Pressure robust finite element methods for incompressible flows provide errors in velocity independent of thepressure error. In this talk we present a pressure robust a posteriori error estimator, which estimates the velocityerror independent of the quality of pressure approximation.

36

Some nonstandard high-efficiency finite element schemesShuo Zhang1

1 LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy ofMathematics and System Sciences, and National Centre for Mathematics and Interdisciplinary Sciences, Chinese

Academy of Sciences, Beijing, China

For practical and theoretical reasons, high-efficiency schemes are of wide interests. The issue of high efficiencymeans more information obtained versus less computational cost. The obtainments include a better approximationfor source and eigenvalue problems and, e.g., a guaranteed upper or lower bound of the eigenvalue, while the costincludes computational resources and time for computation. In this talk, we would like to present some finiteelement schemes which seek to give higher-efficiency performance.

The presented schemes each employs piecewise quadratic polynomials on general quadrilateral grids for thebiharmonic equation, piecewise quadratic polynomials on rectangular grids for the Poisson equation, and piecewisecubic polynomials on general triangulations for biharmonic equation, respectively, and they all exploit the bestpossible accuracy with piecewise polynomials of certain degrees. The schemes fall into the nonconforming finiteelement methods, of the formulation of classical nonconforming ones or of the formulation of DG ones, but aredescribed and constructed in a nonstandard way. Both source and eigenvalue problems will be mentioned, andsome nonstandard phenomena will be demonstrated.

References

[1] S. Zhang, Minimal consistent finite element space for the biharmonic equation on quadrilateral grids, IMA J.Numer. Anal., in press, 2019. DOI:10.1093/imanum/dry096

[2] S. Zhang, Optimal piecewise cubic finite element schemes for the biharmonic equation on general triangula-tions, arXiv:1903.04897, 2019.

[3] H. Zeng, C. Zhang, S. Zhang, Optimal quadratic element on rectangular grids for H1 problems,arXiv:1903.00938, 2019.

37

Parallel implementation of finite element methodsEglantina Kalluci1, Xhilda Merkaj2, and Mentor Shevroja3

1 Department of Applied Mathematics, University of Tirana, Albania2NASRI (National Agency for Scientific Research and Inovation), Albania

3 Department of Applied Mathematics, University of Tirana, Albania

Many of practical problems are modelled using partial differential equations (PDE). In this paper, we introducea problem known as Plasma Source-Ion Implementation (PSII), which is a technique for implanting ions intomaterials to modify their surface properties. The model is one dimensional (planar, cylindrical, or spherical).The time-dependent, self-consistent potential profile is calculated from 1D Poisson’s equation. From the widerange of numerical methods for solving this equation, we have tested various methods and among them the finiteelement method (FEM). We have simulated changes in the surface of an argon material for different values ofpotential using FEM. Furthermore, in this paper, we analyze the parallel implementation of the finite elementsmethods, which is used for solving the above one dimensional Poisson equation. Numerical tests are performedusing OpenMP/ C++ platform. We have performed numerical tests to find the optimal number of threads neededto achieve the best execution time of the algorithm. Also, other performance evaluations discussed are speed-upand effectiveness.

References

[1] C. C. Douglas, G. Haase and U. Langer, A Tutorial on Elliptic PDE Solvers and their Parallelization, SIAMSeries Software, Environments, and Tools, 2003.

[2] G. A. Emmett and M. A. Henry, Numerical simulation of plasma sheath expansion, with applications toplasma-source ion implantation, Journal of Applied Physics 71:113–117, 1992. DOI:10.1063/1.350740

[3] D. Ganellari, G. Haase and G. W. Zumbusch, A massively parallel Eikonal solver on unstructured meshes,G. Comput. Visual Sci. 19(5–6):3–18, 2018. DOI:10.1007/s00791-018-0288-z

38

Convergence analysis of adaptive stochastic Galerkin FEMAlex Bespalov1, Dirk Praetorius2, Leonardo Rocchi1, and Michele Ruggeri2

1School of Mathematics, University of Birmingham, United Kingdom2Institute for Analysis and Scientific Computing, TU Wien, Austria

We study adaptive stochastic Galerkin finite element approximations for a class of elliptic partial differentialequations with parametric or uncertain coefficients. The energy error in Galerkin approximations is controlledby a novel reliable and efficient a posteriori error estimator that combines a two-level spatial estimator and ahierarchical parametric estimator, see [2]. The structure of the estimator is exploited in the proposed algorithmto perform a balanced adaptive refinement of the spatial and parametric components of Galerkin approximations.In this talk, we present recent convergence results for this adaptive refinement procedure [1]. In particular, weshow that the proposed algorithm drives the underlying energy error estimate to zero. We also report the resultsof numerical experiments that (i) illustrate the advantages of adaptive refinement of both (spatial and stochastic)components of Galerkin approximations, and (ii) explore an optimal choice of threshold parameters in differentmarking strategies that can be employed for adaptive refinement.

References

[1] A. Bespalov, D. Praetorius, L. Rocchi, and M. Ruggeri, Convergence of adaptive stochastic Galerkin FEM,SIAM J. Numer. Anal., accepted, 2019. Preprint available at arXiv:1811.09462.

[2] A. Bespalov, D. Praetorius, L. Rocchi, and M. Ruggeri, Goal-oriented error estimation and adaptivity forelliptic PDEs with parametric or uncertain inputs, Comput. Methods Appl. Mech. Engrg. 345:951–982, 2019.DOI:10.1016/j.cma.2018.10.041

39

Instance-optimal goal-oriented adaptivityMichael Innerberger1 and Dirk Praetorius1

1Institute for Analysis and Scientific Computing, TU Wien, Austria

Adaptive finite element methods (AFEM) for elliptic PDEs iterate the loop solve–estimate–mark–refineto successively adapt an initial mesh T0 to the possible singularities of the sought solution u ∈ H1

0 (Ω) in orderto obtain a sequence of meshes T` and corresponding approximate solutions u` ≈ u. While usual strategies forAFEM employ some variant of the Dörfler marking criterion [1] and aim for optimal convergence rates of the error‖∇(u− u`)‖L2(Ω), [2] considers a modified maximum strategy to obtain an instance-optimal AFEM, i.e., the erroron T` is quasi-optimal with respect to all refinements of T0, which have essentially the same number of elements:It holds that

∃C > 1 ∀ ` ∈ N0 ∀ Th ∈ refine(T0) :[C #Th ≤ #T` =⇒ ‖∇(u − u`)‖L2(Ω) ≤ C ‖∇(u − uh)‖L2(Ω)

].

In goal-oriented adaptivity, an additional goal functional G ∈ H−1(Ω) is given. Instead of minimizing only theenergy error ‖∇(u − u`)‖L2(Ω), one aims for

|G (u)− G (uh)| −→ min .

We consider an adaptive finite element method with arbitrary but fixed polynomial degree p ≥ 1, where adaptivityis driven by an edge-based residual error estimator. Based on the modified maximum criterion of [2], we proposea goal-oriented adaptive algorithm and prove that it is instance optimal. We give numerical examples to underlineour findings and compare our algorithm to conventional marking strategies for rate-optimal goal-oriented AFEMs[3].

References

[1] W. Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33(3):1106–1124,1996. DOI:10.1137/0733054,

[2] L. Diening, C. Kreuzer, and R. Stevenson, Instance Optimality of the Adaptive Maximum Strategy, Found.Comput. Math. 16:33–68, 2016. DOI:10.1007/s10208-014-9236-6

[3] M. Feischl, D. Praetorius and G. van der Zee, An abstract analysis of optimal goal-oriented adaptivity, SIAMJ. Numer. Anal. 54(3):1423–1448, 2016. DOI:10.1137/15M1021982

40

Preconditioning techniques applied on optimal control problems anddiscretized by a fictitious domain FEM with cut elements

Aikaterini Aretaki1 and Efthymios Karatzas1,2

1Department of Mathematics, National Technical University of Athens, Greece2mathLab Laboratory, International School for Advanced Studies Trieste, Italy

This presentation focuses on the performance among some fundamental preconditioners applied on the sparselinear system arising from quadratic optimal control problems with Dirichlet boundary conditions and discretized viaa fictitious domain finite element method using cut elements [1]–[5]. We emphasize on block-diagonal structures [6]whose diagonal blocks are scaled or coarse grid approximations of the stiffness matrices coming from the successivediscretization of the state and adjoint variational forms. The properties of symmetry and positive definitenessinherited from the cut finite element method enable us to use these preconditioners together with Krylov subspaceiterative solution methods such as the preconditioned conjugate gradient algorithm. Apart from testing theclassical point Jacobi and the symmetrized Gauss-Seidel preconditioners, alternative block versions are analogouslydefined. Herein, the strategy for constructing the appropriate block variants concentrates on a vertex partition ofthe computational mesh which allows finite elements to straddle between subdomains, that is, they are allowedto overlap. By gathering the overlapping blocks into a block diagonal matrix and employing a symmetrizedGauss-Seidel iteration, a symmetric block-Jacobi preconditioner is guaranteed. Furthermore, combining this blockdiagonal structure with a coarse grid correction scheme, an additive coarse grid preconditioner is constructed.The resulting scheme seems to be optimal for the corresponding state and adjoint discrete matrix systems inthe sense that the number of iterations achieved for convergence appears to be independent of the mesh size.The aforementioned approach also yields the predicted order rate of convergence for the solution of the discreteoptimality system, consistent with the convergence rate of the cut finite element method. Application to a Stokesflow model system of equations is also provided. The quality of the prescribed preconditioners has been verifiednumerically.

References

[1] A. Aretaki, and E. N. Karatzas, Preconditioning techniques applied on optimal control problems and dis-cretized by a fictitious domain FEM with cut elements, in preparation, 2019.

[2] E. Burman, and P. Hansbo, Fictitious domain finite element methods using cut elements II. A stabilizedNitsche method, Appl. Num. Math. 64(4):328–341, 2011. DOI:10.1016/j.apnum.2011.01.008

[3] E. N. Karatzas, F. Ballarin, and G. Rozza, Projection-based reduced order models for a cut finite elementmethod in parametrized domains, arXiv:1901.03846, 2019.

[4] E. N. Karatzas, and G. Katsouleas, Cut finite element method for Dirichlet control problems in semilinearelliptic PDEs, in preparation, 2019.

[5] E. N. Karatzas, G. Stabile, L. Nouveau, G. Scovazzi, and G. Rozza, A reduced basis approach for PDEs onparametrized geometries based on the shifted boundary finite element method and application to a Stokesflow, Comput. Methods Appl. Mech. Engrg. 347:568–587, 2019. DOI:10.1016/j.cma.2018.12.040

[6] T. Rees, H. S. Dollar, and A. J. Wathen, All-at-once preconditioning in pde-constrained optimization, Ky-bernetica 46(2):341–360, 2010. DOI:10.1006/jcph.1999.6368

41

Error analysis of free boundary problemsSergey Repin1,2

1Department of Mathematical Information Technology, University of Jyväskylä, Finland2Steklov Institute of Mathematics, St. Petersburg, Russia

We present a general method that generates error relations for a wide class of nonlinear variational problems.Applications are related to several free boundary problems, such as the classical and two-phase obstacle problems,problems with thin obstacle, and plasticity models.

42

Functional a posteriori error estimatesfor boundary element methods

Stefan Kurz1, Dirk Pauly2, Dirk Praetorius3, Sergey Repin4,5,and Daniel Sebastian3

1Institute for Accelerator Science and Electromagnetic Fields, TU Darmstadt, Germany2Fakultät für Mathematik, Universität Duisburg-Essen, Germany

3Institute for Analysis and Scientific Computing, TU Wien, Austria4Department of Mathematical Information Technology, Jyväskylän Yliopisto, Finland

5Steklov Institute of Mathematics, St. Petersburg, Russia

This work motivates a new perspective on a posteriori error estimation for boundary element methods (BEMs).In order to solve

∆u = 0 in Ω ⊂ Rd , (1)u|Γ = g on Γ = ∂Ω, (2)

we employ an indirect BEM approach to solve Symm’s integral equation

Vφ(x) :=

∫Γ

G (x − y)φ(y) dy = g(x) for almost all x ∈ Γ

and use the BEM approximation φh ≈ φ ∈ H−1/2(Γ ) to obtain

uh(x) :=

∫Γ

G (x − y)φh(y) dy ≈ u(x) for almost all x ∈ Ω.

In contrast to the state of the art, we aim for fully computable lower and upper bounds of the physically relevantpotential error ||∇(u − uh)||L2(Ω). To this end, we develop functional-type estimates in the spirit of [1]. BEM’sdistinguishing feature that the error u − uh solves (1) exactly is the only ingredient to conclude the sharp erroridentity

maxτ∈L2(Ω)divτ=0

[2⟨(g − uh|Γ ) , (n · τ )|Γ

⟩− ||τ ||2L2(Ω)

]= ||∇(u − uh)||2L2(Ω) = min

w∈H1(Ω)w|Γ=g−uh|Γ

||∇w ||2L2(Ω)

which leads to maximization/minimization procedures. Instead of solving the related problems on Ω, we suggesta construction via FEM on an adaptively shrinking boundary layer S ⊂ Ω. Our computations on S are then usedto steer an adaptive mesh-refinement on Γ to regain the optimal rate of convergence

||∇(u − uh)||L2(Ω) ≤ Ccont||φ− φh||H−1/2(Γ ) = O(N−3/2).

The numerical examples verify that the majorant provides a reasonable stopping criterion, in order to guaranteethe integrity of the final approximate solution.

We note that the developed approach covers indirect and direct BEM and does not exploit the Galerkinorthogonality. In particular, it is thus applicable to, e.g., the collocation method.

References

[1] S. Repin, A posteriori error estimates for partial differential equations, Radon Series on Computational andApplied Mathematics 4, Walter de Gruyter, Berlin, 2008.

43

A convergent finite element boundary element scheme forMaxwell-Landau-Lifshitz-Gilbert equations

Jan Bohn1, Michael Feischl2, and Willy Dörfler1

1Institut für Angewandte und Numerische Mathematik, Karlsruhe Institute of Technology, Germany2Institute for Analysis and Scientific Computing, TU Wien, Austria

We consider the Landau-Lifshitz-Gilbert-equation (LLG) on a bounded domain Ω with Lipschitz-boundaryΓ coupled with the linear Maxwell equations on the whole space. As the material parameters outside of Ω areassumed to be constant, we are able to reformulate the problem to a MLLG system in Ω coupled to a boundaryequation on Γ .

We define a suitable weak solution (which still has a reasonable trace for the boundary equation) and proposea time-stepping algorithm which decouples the Maxwell part and the LLG part of the system and which onlyneeds linear solvers even for the nonlinear LLG part. The approximation of the boundary integrals is done withconvolution quadrature.

Under weak assumptions on the initial data and the input parameters we show convergence of the algorithmtowards weak solutions, which especially guarantees the existence of solutions to the MLLG system.

References

[1] L. Banas, M. Page, and D. Praetorius, A convergent linear finite element scheme for the Maxwell-Landau-Lifshitz-Gilbert equations, Electron. Trans. Numer. Anal. 44:250–270, 2015.

[2] A. Buffa, and R. Hiptmair. Galerkin boundary element methods for electromagnetic scattering, SpringerTopics in computational wave propagation 31, 2003. DOI:10.1007/978-3-642-55483-4_3

[3] B. Kovács, and C. Lubich, Stable and convergent fully discrete interior-exterior coupling of Maxwell’s equa-tions, Numer. Math. 137:91–117, 2017. DOI:10.1007/s00211-017-0868-8

[4] C. Lubich, Convolution quadrature revisited, BIT 44:503–514, 2004.DOI:10.1023/B:BITN.0000046813.23911.2d

[5] M. Scroggs, T. Betcke, E. Burman, W. Śmigaj, E. van ’t Wout, Software frameworks for integral equationsin electromagnetic scattering based on Calderón identities. Comput. Math. Appl. 74(11):2897–2914, 2017.DOI:10.1016/j.camwa.2017.07.049

44

DPG theory and techniques for plate bending problemsThomas Führer1, Alexander Haberl2, Norbert Heuer1, Antti Niemi3, and Francisco-Javier Sayas4

1Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Chile2Institute for Analysis and Scientific Computing, Technische Universität Wien, Austria

3Faculty of Technology, University of Oulu, Finland4Department of Mathematical Sciences, University of Delaware, USA

In recent years, the discontinuous Petrov–Galerkin method with optimal test functions (DPG method) hasbeen established as a framework that can deliver automatically discrete inf-sup stable approximations [2, 1]. Thisis particularly useful for singularly perturbed problems.

In this talk we will give a brief introduction to this framework, and discuss its design and analysis for theimportant class of plate bending models. For reasons of flexibility in the selection of variables and for ease ofanalysis, the variational formulation of choice is of ultraweak type. This means that all the appearing derivativesare transferred onto the test side, which is a product space by design. It also means that the regularity of theproblem entirely resides within trace operations and spaces. We discuss these traces for various plate bendingmodels, the Kirchhoff–Love model [5, 4], the bi-Laplacian [3], and the Reissner–Mindlin model [6]. Here, thefocus is on using low regularity assumptions so that the corresponding formulation and scheme are relevant forcritical engineering applications. Specifically, we deal with the Reissner–Mindlin model and its Kirchhoff–Lovelimit in a uniform way, including the case of non-convex polygonal plates. The DPG research on plate problems isby no means complete. We will discuss open problems and objectives for future research.

Financial support by CONICYT-Chile through Fondecyt project 1190009 is gratefully acknowledged.

References

[1] C. Carstensen, L. Demkowicz, and J. Gopalakrishnan, Breaking spaces and forms for the DPGmethod and applications including Maxwell equations, Comput. Math. Appl., 72:494–522, 2016.DOI:10.1016/j.camwa.2016.05.004

[2] L. Demkowicz and J. Gopalakrishnan, A class of discontinuous Petrov-Galerkin methods. Part II: Optimaltest functions, Numer. Methods Partial Differential Eq., 27:70–105, 2011. DOI:10.1002/num.20640

[3] T. Führer, A. Haberl, and N. Heuer, Trace operators of the bi-Laplacian and applications, arXiv:1904.07761,2019.

[4] T. Führer and N. Heuer, Fully discrete DPG methods for the Kirchhoff–Love plate bending model, Comput.Methods Appl. Mech. Engrg., 343:550–571, 2019. DOI:10.1016/j.cma.2018.08.041

[5] T. Führer, N. Heuer, and A. Niemi, An ultraweak formulation of the Kirchhoff–Love plate bending modeland DPG approximation, Math. Comp., 88:1587–1619, 2019. DOI:10.1090/mcom/3381

[6] T. Führer, N. Heuer, and F.-J. Sayas, An ultraweak formulation of the Reissner–Mindlin plate bending modeland DPG approximation, arXiv:1906.04869, 2019.

45

On the nonsymmetric coupling method for parabolic-ellipticinterface problems

Christoph Erath1

1Department of Mathematics, TU Darmstadt, Germany

We consider a parabolic-elliptic interface problem in an unbounded domain. An application is, for instance, themodeling of eddy currents in the magneto-quasistatic regime. Due to different physical properties of the solutionin different parts of the domain it makes sense to consider couplings of different methods to get the best possiblenumerical approximation.

Therefore, we introduce the nonsymmetric coupling of the Finite Element Method and the Boundary ElementMethod to approximate such a solution in space without truncation of the unbounded domain [1, 2]. This leads toa semi-discrete solution. For the subsequent time discretization we either use the classical backward Euler schemeor a variant of it to get a fully-discrete system. The model problem and its numerical discretization are well-posedalso for Lipschitz interfaces [3]. Even more, we prove quasi-optimality, i.e., the best possible approximation ofthe solution, of our semi-discrete and fully-discrete solution under minimal regularity assumptions in the naturalenergy norm [3]. We remark that our analysis does not use duality arguments and corresponding estimates foran elliptic projection which are not available for our nonsymmetric coupling method. Instead, we use estimates inappropriate energy norms. Moreover, we discuss the extension of our model problem to a problem arising in fluidmechanics. However, since the conservation of fluxes is mandatory for such applications, we replace the FiniteElement Method by the Finite Volume Method in our coupling approach [4]. Numerical examples illustrate thepredicted (optimal) convergence rates and underline the potential for practical applications.

References

[1] R. C. MacCamy and M. Suri, A time-dependent interface problem for two-dimensional eddy currents, Quart.Appl. Math., 44:675–690, 1987.

[2] F.-J. Sayas, The validity of Johnson-Nédélec’s BEM-FEM coupling on polygonal interfaces, SIAM J. Numer.Anal. 47(5): 3451–3463, 2018. DOI:10.1137/08072334X

[3] H. Egger, C. Erath, and R. Schorr, On the nonsymmetric coupling method for parabolic-elliptic interfaceproblems, SIAM J. Numer. Anal. 56(6): 3510–3533, 2018. DOI:10.1137/17M1158276

[4] C. Erath and R. Schorr, Stable non-symmetric coupling of the finite volume method and the boundary ele-ment method for convection-dominated parabolic-elliptic interface problems, Comput. Methods Appl. Math.published online, 2019. DOI: 10.1515/cmam-2018-0253

46

A finite volume scheme preserving maximum principle for thesystem of radiation diffusion equations with three-temperature

Yunlong Yu1 and Guangwei Yuan1

1Institute of Applied Physics and Computational Mathematics, Beijing, China

Radiation transport in astrophysical phenomena and inertial confinement fusion (ICF) is often modeled by thediffusion approximation. When the radiation field is not in the thermodynamic equilibrium, a coupled system ofnon-equilibrium diffusion equations is used to simulate the radiation transport, including both radiation diffusionand heat conduction in the material. These non-equilibrium equations are highly nonlinear, tightly coupled, andexhibit multiple time and space scales. So, it is a very challenging problem to construct a numerical algorithmwith high accuracy and efficiency.

The maximum principle is an important feature of the equilibrium diffusion equation. Under some additionalnon-negativity assumptions on the exchange coefficients, it is also satisfied by the non-equilibrium diffusion equa-tions. As we know, the discrete maximum principle (DMP) ensures the nonnegativity and eliminates non-physicaloscillations of numerical solutions. Hence, it is a desirable property for the numerical schemes.

Many diffusion schemes preserving DMP are constructed in the past decades [1]-[3]. However, most of themare focused on solving the single equation. We are interesting in designing a scheme preserving DMP for thefollowing the non-equilibrium three-temperature equations

ρcve∂ue

∂t−∇ · (κe∇ue) = ωei (ui − ue) + ωer (ur − ue) + We , in Ω\Γ , (1)

ρcvi∂ui

∂t−∇ · (κi∇ui ) = ωei (ue − ui ) + Wi , in Ω\Γ , (2)

ρcvr∂ur

∂t−∇ · (κr∇ur ) = ωer (ue − ur ), in Ω\Γ , (3)

where t > 0 and Ω is a bounded polygonal domain in R2, ρ is the material density, Γ is the material interface,u = (ue , ui , ur ) denotes the electron, ion and photon temperatures, respectively.

We propose a cell-centered nonlinear finite volume scheme for the above equations, where both the Dirichletand Neumann boundary conditions are considered, and prove that the discrete solutions of the scheme satisfy thediscrete maximum principle. In the construction of the flux we use the nonlinear combination of two-single fluxesand the interpolation technique for the auxiliary unknowns. Some new interpolation methods are introduced,especially when the Neumann boundary condition is considered. Based on the bounded estimation of the discretesolution, we prove that there exists at least a solution for our scheme by using the Brouwer’s fix point theorem.Numerical results show that our scheme has second-ordered accuracy, good conservation and can preserve thediscrete maximum principle.

References

[1] E. Bertolazzi, G. Manzini, A second-order maximum principle preserving finite volume method for steadyconvection-diffusion problems, SIAM J. Numer. Anal. 43:2172–2199, 2015. DOI:10.1137/040607071

[2] J. Droniou, C. Le Potier, Construction and convergence study of schemes preserving the elliptic local maximumprinciple, SIAM J. Numer. Anal. 49:459–490, 2011. DOI:10.1137/090770849

[3] G. Yuan, Y. Yu, Existence of solution of a finite volume scheme preserving maximum principle for diffusionequations, Numer. Methods Partial Diff. Eqs. 34(1):89–96, 2018. DOI:10.1002/num.22184

47

Numerical methods for viscoplastic models in metal formingprocesses

Ioannis Toulopoulos1,2

1AC2T research GmbH, Austrian Excellence Center for Tribology, Austria2Institute of Computational Mathematics, Johannes Kepler University Linz, Austria

The mathematical modeling and the numerical simulation of rolling metal processes, i.e., the plastic deformationof a metal plate passing through a pair of rotating rolls, are subjects of investigation of many researchers. In thiswork we consider a rigid-viscoplastic model (Norotn-Hoff power law model) for describing a plane strain rollingprocess and develop a continuous finite element scheme for its discretization. Along the common interface, theunilateral contact conditions are described by inequality constraints and the stick/slip motion along the interfacebetween the roll and the plate is described by extending the usual Coulomb’s law to viscoplastic frame.

We present the finite element discretization of the resulting equilibrium equations and analyse the numericaltreatment of the contact and slipping conditions across the common interface. The associated contact conditionsinclude the normal component terms of the relative velocity are incorporated in the finite element scheme byusing Nitsche-type (penalty) techniques. The tangential motion conditions (frictional sliding motion) includerelative velocity terms and are weakly imposed in the numerical scheme by using appropriate numerical fluxes.The discontinuous nature of this approach gives several advantages for the case of having general non-matchinginterface meshes. Numerical results related to the variation of the important variables, e.g., velocity components,normal stress, etc, after long time integration are discussed. The results presented in this talk are based on [1],[2]. This work is supported by the project COMET K2 XTribology, No 849109, Project grantee: Excellence Centerfor Tribology, and by the project “JKU-LIT-2017-4-SEE-004”.

References

[1] I. Toulopoulos, M. Jech, G. Verllaufer, Viscoplastic models and finite element schemes for rolling process,in preparation, 2019.

[2] S. Nakov, I. Toulopoulos, A space time finite element scheme for nonlinear parabolic problems, in preparation,2019.

[3] S. Nakov, I. Toulopoulos, An adaptive mesh refinment for nonlinear problems problems, in preparation, 2019.

48

Layer-adapted piecewise uniform Shishkin-type meshes for solvingboundary problems of the singular perturbated fourth-order

differential equationIrina Kolosova1 and Sergey Vasilyev1

1 Department of Applied Probability and Informatics, RUDN University, Russia

In the papers [1, 2], a relativistic analogue of the Schrödinger equation was considered and the asymptoticsolution of the boundary value problem for this singular perturbated infinite order differential equation was obtained.We can use a truncation method to eliminate high order derivatives in this equation and consider the boundaryproblems for the singular perturbated fourth-order differential equation of the form

εd2

dx2

[k(x , ε)

d2u

dx2

]− d

dx

[p(x , ε)

du

dx

]+ q(x , ε)u = f (x , ε),

with the boundary conditions A

u(0) = 0, u(1) = 0, u′(0) = 0, u′(1) = 0,

or the boundary conditions Bu(0) = 0, u(1) = 0, u′′(0) = 0, u′′(1) = 0,

where k(x , ε), p(x , ε), q(x , ε), f (x , ε) are some continuous functions of all variables, and we suppose that

0 < c1 ≤ k(x , ε) ≤ c2, 0 ≤ p(x , ε) ≤ c3, 0 ≤ q(x , ε) ≤ c4, |f (x , ε)| < c5,

where ci > 0, i = 1, 5 are some positive constants, and ε > 0, ε << 1 is a small parameter.In this talk, we propose modifications of the layer-adapted piecewise uniform Shishkin-type meshes [3, 4, 5]

for numerical solving problems A and B. We show that this modified Shishkin-type methods for the numericalsolution of the boundary problems A and B is uniformly convergent, in the discrete maximum norm, independentof singular perturbation parameter.

References

[1] I. Amirkhanov, E. Zhidkov, I. Zhidkova, and S. Vasilyev, Construction of an asymptotic approximation ofeigenfunctions and eigenvalues of a boundary value problem for the singular perturbed relativistic analog ofthe Schrödinger equation with an arbitrary potential, Mat. Model. 15(9):3–16, 2003.

[2] I. Amirkhanov, S. Vasilyev, D. Vasilyeva, A. Karaschiuk, V. Denisov, and D. Udin, Asymptotic solution ofboundary problem for relativistic finite-difference Schrödinger equation with singular oscillator quasipotential,Bulletin of PFUR, series ”Mathematics. Informatics. Physics.” 3:55–68, 2008.

[3] V. Andreev, and I. Savin, The uniform convergence with respect to a small parameter of A. A. Samarskii’smonotone scheme and its modification, Computational Mathematics and Mathematical Physics. 35:581–591,1995.

[4] T. Linss, H.-G. Roos, and R. Vulanovic, Uniform pointwise convergence on Shishkin-type meshes for quasi-linear convection-diffusion problems, SIAM Journal on Numerical Analysis. 38(3):897–912, 2000.

[5] R. Vulanovic, and L. Teofanov A modification of the Shishkin discretization mesh for one-dimensionalreaction–diffusion problems, Appl. Math. Comput. 220:104–116, 2013. DOI:10.1016/j.amc.2013.05.055

49

A meta-heuristic crow search algorithm for solving the open vehiclerouting problem with time windows

Fateme Maleki1, Majid Yousefikhoshbakht2, and Andrea D′Ariano3

1Department of Computer Science, Faculty of Mathematics,University of Sistan and Baluchestan, Zahedan, Iran

2Department of Mathematics, Faculty of Sciences, Bu-Ali Sina University, Hamedan, Iran,3Department of Engineering, Section of Computer Science and Automation,

Roma Tre University, Rome, Italy

The open vehicle routing problem with time windows (OVRPTW) is one of the well-known routing problemswith many real-world applications, such as rail, bus and air transportation planning. In this paper, we propose ahybrid crow search algorithm (HCSA) to effectively solve the OVRPTW as a discrete problem. In the proposedHCSA, the hill climbing strategy is applied as a local search algorithm with three local search neighborhoodstructures. HCSA also uses the elitism approach to select the initial solutions of the OVRPTW. The effectivenessof the HCSA is demonstrated on two benchmark sets available in the literature: a small-scale benchmark ofnumerical examples and the Solomon’s benchmark instances. The results of the comparison confirm that HCSAproduces very competitive results with respect to other published methods. From a set of 56 Solomon’s benchmarkinstances, HCSA computes 5 new best-known solutions and finds the best-known solution for more than 50% ofthe instances.

References

[1] A. Askarzadeh, A novel metaheuristic method for solving constrained engineering optimization problems:crow search algorithm, Computers & Structures 169:1–12, 2016. DOI:10.1016/j.compstruc.2016.03.001

[2] J.C. Holzhaider, G.R. Hunt, R.D. Gray, Social learning in New Caledonian crows, Learning & Behavior38(3):206–219, 2010. DOI:10.3758/LB.38.3.206

[3] J. Brandão, Iterated local search algorithm with ejection chains for the open vehicle routing problem withtime windows, Computers & Industrial Engineering 120:146–159, 2018. DOI:10.1016/j.cie.2018.04.032

[4] X. Zou, L. Liu, K. Li, W. Li, A coordinated algorithm for integrated production schedulingand vehicle routing problem, International Journal of Production Research 56(15):5005–5024, 2018.DOI:10.1080/00207543.2017.1378955

50

hp-FEM for variational inequalities of the second kind involving thep-Laplacian

Lothar Banz1 and Ernst P. Stephan2

1Department of Mathematics, University of Salzburg, Austria2Institute of Applied Mathematics, Leibniz University Hannover, Germany

We consider variational inequalities of the second kind in which the differential operator corresponds to the p-Laplacian and the friction functional is of the type

∫g |v |dx or

∫g |∇v |dx . These kind of problems are discretized

by hp-finite elements. An additional discretization of the friction functional seems to be beneficial for the iterativesolver and the implementation itself. We analyze the approximation scheme in terms of well-posedness and apriori error estimates. Moreover, we derive a family of reliable a posteriori error estimators, of which a computablemember is in some sense also efficient. Adaptivity allows to improve the order of convergence.

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A sparse spectral method for Volterra integral equations usingorthogonal polynomials on the triangle

Timon S. Gutleb1,2 and Sheehan Olver1

1Department of Mathematics, Imperial College London, UK2Department of Mathematics, University of Vienna, Austria

We discuss and introduce a new method in [1] which can numerically solve Volterra integral equations

VK u = f or (λI + VK )u = f

of first and second kind for general kernels K (x , y) with exponential convergence using bivariate orthogonalpolynomials on a triangle domain, building on previous work on bivariate orthogonal polynomials on triangledomains in [2]. The Volterra integral operator

(VK u)(x) :=

∫ x

0

K (x , y)u(y)dy

is shown to be sparse on a weighted triangle Jacobi polynomial basis which allows us to exploit symmetries toachieve high efficiency. We will further discuss proofs for the convergence of the method for general (not necessarilyconvolution) kernels based on connections to the theory of infinite-dimensional Fredholm and Toeplitz operatorsacting on appropriate coefficient sequence spaces. We conclude by showcasing numerical experiments for problemswith known analytic solutions as well as problems with numerically challenging non-convolution oscillatory kernels(Figure 3) without known analytic solutions.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

x

y

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

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1.0

x

y

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

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1.0

x

y

-2.5

-2.0

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

Figure 3: Contour plots of some oscillatory kernels K (x , y) on triangle domains.

References

[1] T. S. Gutleb, S. Olver, A sparse spectral method for Volterra integral equations using orthogonal polynomialson the triangle, arXiv:1906.03907, 2019.

[2] S. Olver, A. Townsend, G. Vasil, A sparse spectral method on triangles, arXiv:1902.04863, 2019

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An efficient explicit scheme for a fourth order nonlinear diffusionfilter

Mahipal Jetta1

1 School of Natural Sciences, Mahindra École Centrale, Hyperabad, India

The existing numerical schemes for multiplicative noise removal fourth order nonlinear diffusion filters requireeither very small time step sizes for stability or involve solving couple of systems of equations. In this work, wedevelop a new explicit scheme using the least square approximation for second order spatial derivatives. We showthat the proposed scheme offers three to four times bigger time step size than the standard finite difference scheme.We further improve the efficiency of this novel scheme with fast explicit diffusion and show through numericalexperiments that the hybrid scheme is more efficient than the alternating direction implicit schemes.

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Recovery of conformity: quasi-optimal and pressure robustdiscretisations of the Stokes equations

Christian Kreuzer1

1Faculty of Mathematics, TU Dortmund, Germany

Nonconforming discretisations often are more flexible than conforming ones and have advantageous conser-vation or stability properties. Unfortunately, their application often requires more regular data and a quasi-bestapproximation property typically holds only up to consistency terms. We present how quasi-optimality can be re-covered under minimal regularity assumptions in a rather general framework. Our construction can be summarisedas follows: First, a linear operator acts on discrete velocity test functions, before the application of the loadfunctional, and maps into the space of conforming functions. Second, we employ a new augmented Lagrangianformulation, inspired by Discontinuous Galerkin methods to preserve consistency.

As an application, we interpret various inf-sup stable non-divergence free Stokes elements as non-conformingdiscretisations in the sense of the divergence constraint, i.e., the pressure error is interpreted as consistency error.The main attention is spend on the linear operator, which needs to map the discrete divergence kernel into thecontinuous one. Its construction employs the solution of local Stokes problems with Scott–Vogelius pairs on anAlfeld sub-triangulation of single elements.

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Computation of the LBB constant with a least-squares finiteelement method

Johannes Storn1

1Department of Mathematics, Bielefeld University, Germany

An investigation of the least-squares finite element method (LSFEM) from [1] for the pseudostress-velocityformulation of the Stokes problem reveals a relation of the Ladyzhenskaya-Babuška-Brezzi (LBB) constant andthe ellipticity constants of the LSFEM. While the approximation of the LBB constant with numerical methods ischallenging (see for example the numerical schemes in [2, 3]), the approximation of the ellipticity constants is in aRayleigh-Ritz-like environment. This setting is well-understood and so allows for an easy to implement convergentnumerical scheme for the computation of the LBB constant. Numerical experiments with uniform and adaptivemesh refinements complement the investigation of this novel scheme.

References

[1] Z. Cai, B. Lee, and P. Wang, Least-squares methods for incompressible Newtonian fluid flow: Linear stationaryproblems, SIAM J. Numer. Anal. 42:843–859, 2004. DOI:10.1137/S0036142903422673

[2] M. Costabel, M. Crouzeix, M. Dauge, and Y. Lafranche, The inf-sup constant for the divergence on cornerdomains, Numer. Methods Partial Differential Equations 31:439–458, 2015. DOI:10.1002/num.21916

[3] D. Gallistl, Rayleigh-Ritz approximation of the inf-sup constant for the divergence, Math. Comp. 88:73–89,2019. DOI:10.1090/mcom/3327

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Optimal convergence rates in L2 for a first order system leastsquares finite element method

Maximilian Bernkopf1 and Jens Markus Melenk1

1Institute for Analysis and Scientific Computing, TU Wien, Austria

We consider a Poisson-like second order model problem written as a system of first order equations. For thediscretization an HHH(Ω, div)×H1(Ω)-conforming least squares formulation is employed. A least squares formulationhas the major advantage that regardless of the original formulation the linear system resulting from a least squarestype discretization is always positive semi-definite, making it easier to solve. Even though our model problem inits standard H1(Ω) formulation is coercive our methods and lines of proof can most certainly be applied to otherproblems as well, see [2, 3] for an application to the Helmholtz equation. A major drawback of a least squaresformulation is that the energy norm is somewhat intractable. Deducing error estimates in other norms, e.g.,the L2(Ω) norm of the scalar variable, is more difficult. Numerical examples in our previous work [2] suggestedconvergence rates previous results did not cover. Closing this gap in the literature will be the main focus of thetalk. To that end we showcase a duality argument in order to derive L2 error estimates of the scalar variable, whichwas the best available estimate in the literature. We then perform a more detailed analysis of the correspondingerror terms. This analysis then leads to optimal convergence rates of the method. The above procedure can thenbe applied to more complicated boundary conditions, for which an analogous result is a nontrivial task. As atool, which is of independent interest, we develop HHH(Ω, div)-conforming approximation operators satisfying certainorthogonality relations. For the analysis, a crucial tool are recently developed projection based commuting diagramoperators, see [4].

References

[1] M. Bernkopf and J.M. Melenk, Optimal convergence rates in L2 for a first order system least squares finiteelement method, in preparation, 2019.

[2] M. Bernkopf and J.M. Melenk, Analysis of the hp-version of a first order system least squares method forthe Helmholtz equation, arXiv:1808.07825, 2018.

[3] H. Chen and W. Qiu A first order system least squares method for the Helmholtz equation, J. Comput. Appl.Math. 309:145–162, 2017. DOI:10.1016/j.cam.2016.06.019

[4] J.M. Melenk and C. Rojik, On commuting p-version projection-based interpolation on tetrahedra,arXiv:1802.00197, 2018.

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LIST OF PARTICIPANTS

1. Aikaterini AretakiNational Technical University of Athens, Greece

2. Lubomir BanasBielefeld University, Germany

3. Sören BartelsUniversity of Freiburg, Germany

4. Alex BespalovUniversity of Birmingham, UK

5. Maximilian BernkopfTU Wien, Austria

6. Jan BohnKarlsruhe Institute of Technology, Germany

7. Iain BuchananUniversity of Strathclyde, UK

8. Carsten CarstensenHumboldt-Universität zu Berlin, Germany

9. Ronan CrannyONERA Toulouse, France

10. Giovanni Di FrattaTU Wien, Austria

11. Paolo Di StolfoUniversity of Salzburg, Austria

12. Zhaonan DongFORTH, Greece

13. Willy DörflerKarlsruhe Institute of Technology, Germany

14. Herbert EggerTU Darmstadt, Germany

15. Bernhard EndtmayerRICAM Austrian Academy of Sciences, Austria

16. Christoph ErathTU Darmstadt, Germany

17. Sebastian ErtelTU Wien, Austria

18. Markus FaustmannTU Wien, Austria

19. Thomas FührerPontifical Catholic University of Chile, Chile

20. Gregor GantnerTU Wien, Austria

21. Joscha GedickeUniversity of Vienna, Austria

22. Sjoerd GeeversUniversity of Vienna, Austria

23. Timon S. GutlebImperial College London, UK

24. Xudeng HangInstitute of Applied Physics and ComputationalMathematics, China

25. Helmut HarbrechtUniversity of Basel, Switzerland

26. Xing-Shi HeXi’an Polytechnic University, China

27. Pascal HeidUniversity of Bern, Switzerland

28. Norbert HeuerPontifical Catholic University of Chile, Chile

29. Adrian HirnEsslingen University, Germany

30. Michael InnerbergerTU Wien, Austria

31. Mahipal JettaMahindra École Centrale, India

32. Eglantina KalluciUniversity of Tirana, Albania

33. Michael KarkulikUniversidad Técnica Federico Santa María, Chile

34. Christian KreuzerTU Dortmund, Germany

35. Evgeny MalkovichSobolev Institute of Mathematics, Russia

36. Markus MelenkTU Wien, Austria

37. Ornela MulitaSISSA, Italy

38. Ilaria PerugiaUniversity of Vienna, Austria

39. Marco PicassoEPFL, Switzerland

40. Carl-Martin PfeilerTU Wien, Austria

41. Dirk PraetoriusTU Wien, Austria

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42. Sergey I. RepinSteklov Mathematical Institute, Russia

43. Alexander RiederTU Wien, Austria

44. Michele RuggeriTU Wien, Austria

45. Stefan SauterUniversity of Zurich, Switzerland

46. Stefan SchimankoTU Wien, Austria

47. Joachim SchöberlTU Wien, Austria

48. Daniel SebastianTU Wien, Austria

49. Ernst Peter StephanLeibniz University Hannover, Germany

50. Rob StevensonUniversity of Amsterdam, Netherlands

51. Jakub StocekHeriot-Watt University, UK

52. Paul StockerUniversity of Vienna, Austria

53. Johannes StornBielefeld University, Germany

54. Ioannis ToulopoulosAC2T research GmbH & JKU Linz, Austria

55. Sergey VasilyevRUDN University, Russia

56. Raymond van VenetiëUniversity of Amsterdam, Netherlands

57. Michele VisinoniUniversity of Milano-Bicocca, Italy

58. Martin VohralíkInria Paris, France

59. Jan WesterdiepUniversity of Amsterdam, Netherlands

60. Majid YousefikhoshbakhtBu-Ali Sina University, Iran

61. Xin-She YangMiddlesex University, UK

62. Yunlong YuInstitute of Applied Physics and ComputationalMathematics, China

63. Marco ZankUniversity of Vienna, Austria

64. Shuo ZhangChinese Academy of Sciences, China

58