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A Dirichlet-to-Neumann (DtN)Multigrid Algorithm for

Locally Conservative Methods

Sandia National Laboratories is a multi program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000. .

Mary F. WheelerThe University of Texas at Austin – ICES

Tim WildeySandia National Labs

SIAM Conference Computational and Mathematical Issues in the Geosciences

March 21-24, 2011 Long Beach, CA

Motivation: Multinumerics

Coupling of mixed and DG using mortars – G. Pencheva

Local grid refinement around wells

Advantages in using weak coupling (mortars)

Motivation: Multinumerics

Motivation: General Framework

Both MFEM and DG are locally conservative.

Multiscale mortar domain decomposition methods:• Arbogast, Pencheva, Wheeler, Yotov 2007• Girault, Sun, Wheeler, Yotov 2008

General a posteriori error estimation framework: • Vohralik 2007, 2008• Ern, Vohralik 2009, 2010• Pencheva, Vohralik, Wheeler, Wildey 2010

Is there a multilevel solver applicable to both MFEM and DG?

Can it be applied to the case of multinumerics?

Can it be used for other locally conservative methods?

Outline

I. Interface Lagrange Multipliers – Face Centered Schemes

II. A Multilevel Algorithm

III. Multigrid Formulation

IV. Applications

V. Conclusions and Future Work

Mixed methods yield linear systems of the form:

Hybridization of Mixed Methods

Mixed methods yield linear systems of the form:

Hybridization of Mixed Methods

Introduce Lagrange multipliers on the element boundaries:

Hybridization of Mixed Methods

Introduce Lagrange multipliers on the element boundaries:

Hybridization of Mixed Methods

Reduce to Schur complement for Lagrange multipliers:

Hybridization of Mixed Methods

Existing Multilevel Algorithms

Mathematical Formulation

Mathematical Formulation

Assumptions on Local DtN Maps

Defining Coarse Grid Operators

X

A Multilevel Algorithm

A Multilevel Direct Solver

Given a face-centered scheme

A Multilevel Direct Solver

Given a face-centered scheme1. Identify interior DOF

A Multilevel Direct Solver

Given a face-centered scheme1. Identify interior DOF

Eliminate

A Multilevel Direct Solver

Given a face-centered scheme1. Identify interior DOF

Eliminate 2. Identify new interior DOF

A Multilevel Direct Solver

Given a face-centered scheme1. Identify interior DOF

Eliminate 2. Identify new interior DOF

Eliminate

Continue …

Advantages:Only involves Lagrange multipliersNo upscaling of parametersApplicable to hybridized formulations as well as multinumericsCan be performed on unstructured gridsEasily implemented in parallel

Disadvantage: Leads to dense matrices

A Multilevel Direct Solver

An Alternative Multilevel Algorithm

Given a face-centered scheme

Given a face-centered scheme1. Identify interior DOF

An Alternative Multilevel Algorithm

An Alternative Multilevel Algorithm

Given a face-centered scheme1. Identify interior DOF

Coarsen

An Alternative Multilevel Algorithm

Given a face-centered scheme1. Identify interior DOF

Coarsen Eliminate

An Alternative Multilevel Algorithm

Given a face-centered scheme1. Identify interior DOF

Coarsen Eliminate

2. Identify new interior DOF

An Alternative Multilevel Algorithm

Given a face-centered scheme1. Identify interior DOF

Coarsen Eliminate

2. Identify new interior DOF Coarsen

An Alternative Multilevel Algorithm

Given a face-centered scheme1. Identify interior DOF

Coarsen Eliminate

2. Identify new interior DOF Coarsen Eliminate

Continue …

How to use these coarse level operators?

An Alternative Multilevel Algorithm

Multigrid Formulation

A Multigrid Algorithm

A Multigrid Algorithm

A Multigrid Algorithm

A Multigrid Algorithm

A Multigrid Algorithm

A Multigrid Algorithm

Theorem

A Multigrid Algorithm

Numerical Results

Laplace Equation - Mixed

Levels DOF V-cycles MG Factor

3 224 8 0.194 960 8 0.225 3968 9 0.236 16128 9 0.247 65024 9 0.24

Laplace Equation – Symmetric DG

Levels DOF V-cycles MG Factor

3 224 8 0.204 960 8 0.215 3968 8 0.216 16128 8 0.217 65024 8 0.21

Laplace Equation – Symmetric DG

Levels DOF V-cycles MG Factor

3 224 5 0.084 960 5 0.085 3968 5 0.086 16128 5 0.087 65024 5 0.08

Laplace Equation – Nonsymmetric DG

Levels DOF V-cycles MG Factor

3 224 7 0.164 960 7 0.175 3968 7 0.176 16128 7 0.177 65024 7 0.17

Laplace Equation – Nonsymmetric DG

Levels DOF V-cycles MG Factor

3 224 8 0.184 960 8 0.185 3968 8 0.196 16128 8 0.197 65024 8 0.19

Laplace Equation – Multinumerics

Laplace Equation – Multinumerics

Levels DOF V-cycles MG Factor

3 224 8 0.194 960 8 0.195 3968 8 0.206 16128 8 0.20

Advection - Diffusion

Levels DOF V-cycles MG Factor PGMRES Iters

4 960 10 0.23 75 3968 7 0.11 66 16128 8 0.11 57 65024 9 0.14 5

Poisson Equation – Unstructured Mesh

Single Phase Flow with Heterogeneities

Conclusions and Future Work

Developed an optimal multigrid algorithm for mixed, DG, and multinumerics.

No subgrid physics required on coarse grids only local Dirichlet to Neumann maps.

No upscaling of parameters. Only requires solving local problems (of flexible size). Applicable to unstructured meshes. Physics-based projection and restriction operators. Extends easily to systems of equations (smoothers?)? Analysis for nonsymmetric operators/formulations? Algebraic approximation of parameterization

Thank you for your attention!Questions?

Poisson Equation - Full Tensor

Levels DOF V-cycles MG Factor PCG Iters

3 224 13 0.36 74 960 17 0.46 85 3968 19 0.49 96 16128 20 0.48 107 65024 21 0.47 10

Poisson Equation - Jumps in Permeability

Levels DOF V-cycles MG Factor PGMRES Iters

3 224 15 0.35 104 960 31 0.61 85 3968 29 0.59 86 16128 27 0.56 87 65024 25 0.52 8