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-AC04-94AL Mary F. Wheeler The University of Texas at Austin – ICES Tim Wildey Sandia 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 12

13

Assumptions on Local DtN Maps 14

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 scheme 1.Identify interior DOF

A Multilevel Direct Solver Given a face-centered scheme 1.Identify interior DOF  Eliminate

A Multilevel Direct Solver Given a face-centered scheme 1.Identify interior DOF  Eliminate 2.Identify new interior DOF

A Multilevel Direct Solver Given a face-centered scheme 1.Identify interior DOF  Eliminate 2.Identify new interior DOF  Eliminate Continue …

Advantages: Only involves Lagrange multipliers No upscaling of parameters Applicable to hybridized formulations as well as multinumerics Can be performed on unstructured grids Easily implemented in parallel Disadvantage: Leads to dense matrices A Multilevel Direct Solver

An Alternative Multilevel Algorithm Given a face-centered scheme

1.Identify interior DOF An Alternative Multilevel Algorithm

Given a face-centered scheme 1.Identify interior DOF  Coarsen

An Alternative Multilevel Algorithm Given a face-centered scheme 1.Identify interior DOF  Coarsen  Eliminate

An Alternative Multilevel Algorithm Given a face-centered scheme 1.Identify interior DOF  Coarsen  Eliminate 2.Identify new interior DOF

An Alternative Multilevel Algorithm Given a face-centered scheme 1.Identify interior DOF  Coarsen  Eliminate 2.Identify new interior DOF  Coarsen

An Alternative Multilevel Algorithm Given a face-centered scheme 1.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

Theorem A Multigrid Algorithm

Numerical Results

Laplace Equation - Mixed LevelsDOFV-cyclesMG Factor

Laplace Equation – Symmetric DG LevelsDOFV-cyclesMG Factor

Laplace Equation – Symmetric DG LevelsDOFV-cyclesMG Factor

Laplace Equation – Nonsymmetric DG LevelsDOFV-cyclesMG Factor

Laplace Equation – Nonsymmetric DG LevelsDOFV-cyclesMG Factor

Laplace Equation – Multinumerics

LevelsDOFV-cyclesMG Factor

Advection - Diffusion LevelsDOFV-cyclesMG FactorPGMRES Iters

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 LevelsDOFV-cyclesMG FactorPCG Iters

Poisson Equation - Jumps in Permeability LevelsDOFV-cyclesMG FactorPGMRES Iters