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1
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
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Motivation: Multinumerics
Coupling of mixed and DG using mortars – G. Pencheva
Local grid refinement around wells
Advantages in using weak coupling (mortars)
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Motivation: Multinumerics
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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?
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Outline
I. Interface Lagrange Multipliers – Face Centered Schemes
II. A Multilevel Algorithm
III. Multigrid Formulation
IV. Applications
V. Conclusions and Future Work
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Mixed methods yield linear systems of the form:
Hybridization of Mixed Methods
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Mixed methods yield linear systems of the form:
Hybridization of Mixed Methods
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Introduce Lagrange multipliers on the element boundaries:
Hybridization of Mixed Methods
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Introduce Lagrange multipliers on the element boundaries:
Hybridization of Mixed Methods
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Reduce to Schur complement for Lagrange multipliers:
Hybridization of Mixed Methods
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Existing Multilevel Algorithms
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Mathematical Formulation
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Mathematical Formulation
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Assumptions on Local DtN Maps
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Defining Coarse Grid Operators
X
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A Multilevel Algorithm
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A Multilevel Direct Solver
Given a face-centered scheme
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A Multilevel Direct Solver
Given a face-centered scheme1. Identify interior DOF
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A Multilevel Direct Solver
Given a face-centered scheme1. Identify interior DOF
Eliminate
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A Multilevel Direct Solver
Given a face-centered scheme1. Identify interior DOF
Eliminate 2. Identify new interior DOF
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A Multilevel Direct Solver
Given a face-centered scheme1. Identify interior DOF
Eliminate 2. Identify new interior DOF
Eliminate
Continue …
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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
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An Alternative Multilevel Algorithm
Given a face-centered scheme
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Given a face-centered scheme1. Identify interior DOF
An Alternative Multilevel Algorithm
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An Alternative Multilevel Algorithm
Given a face-centered scheme1. Identify interior DOF
Coarsen
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An Alternative Multilevel Algorithm
Given a face-centered scheme1. Identify interior DOF
Coarsen Eliminate
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An Alternative Multilevel Algorithm
Given a face-centered scheme1. Identify interior DOF
Coarsen Eliminate
2. Identify new interior DOF
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An Alternative Multilevel Algorithm
Given a face-centered scheme1. Identify interior DOF
Coarsen Eliminate
2. Identify new interior DOF Coarsen
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An Alternative Multilevel Algorithm
Given a face-centered scheme1. Identify interior DOF
Coarsen Eliminate
2. Identify new interior DOF Coarsen Eliminate
Continue …
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How to use these coarse level operators?
An Alternative Multilevel Algorithm
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Multigrid Formulation
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A Multigrid Algorithm
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A Multigrid Algorithm
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A Multigrid Algorithm
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A Multigrid Algorithm
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A Multigrid Algorithm
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A Multigrid Algorithm
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Theorem
A Multigrid Algorithm
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Numerical Results
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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
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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
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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
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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
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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
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Laplace Equation – Multinumerics
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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
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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
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Poisson Equation – Unstructured Mesh
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Single Phase Flow with Heterogeneities
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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
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Thank you for your attention!Questions?
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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
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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