Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps

Robust Coarse Spaces for Systems of PDEs via Generalized Eigenproblems in the Overlaps Nicole Spillane Frédérik Nataf Laboratoire Jacques-Louis Loins, CNRS UMR 7598 Université Pierre et Marie Curie, 75005 Paris, France Victorita Dolean Laboratoire J.-A. Dieudonné, CNRS UMR 6621 Université de Nice-Sophia Antipolis, 06108 Nice Cedex 02, France Patrice Hauret Centre de Technologie de Ladoux, Manufacture des Pneumatiques Michelin 63040 Clermont-Ferrand, Cedex 09, France Clemens Pechstein Institute of Computational Mathematics, Johannes Kepler University Altenberger Str. 69, 4040 Linz, Austria Robert Scheichl Department of Mathematical Sciences, University of Bath, Bath BA27AY, UK NuMa-Report No. 2011-07 November 2011 A 4040 LINZ, Altenbergerstraÿe 69, Austria Technical Reports before 1998: 1995 95-1 Hedwig Brandstetter Was ist neu in Fortran 90? March 1995 95-2 G. Haase, B. Heise, M. Kuhn, U. Langer Adaptive Domain Decomposition Methods for Finite and Boundary Element Equations. August 1995 95-3 Joachim Schöberl An Automatic Mesh Generator Using Geometric Rules for Two and Three Space Dimensions. August 1995 1996 96-1 Ferdinand Kickinger Automatic Mesh Generation for 3D Objects. February 1996 96-2 Mario Goppold, Gundolf Haase, Bodo Heise und Michael Kuhn Preprocessing in BE/FE Domain Decomposition Methods. February 1996 96-3 Bodo Heise A Mixed Variational Formulation for 3D Magnetostatics and its Finite Element Discretisation. February 1996 96-4 Bodo Heise und Michael Jung Robust Parallel Newton-Multilevel Methods. February 1996 96-5 Ferdinand Kickinger Algebraic Multigrid for Discrete Elliptic Second Order Problems. February 1996 96-6 Bodo Heise A Mixed Variational Formulation for 3D Magnetostatics and its Finite Element Discretisation. May 1996 96-7 Michael Kuhn Benchmarking for Boundary Element Methods. June 1996 1997 97-1 Bodo Heise, Michael Kuhn and Ulrich Langer A Mixed Variational Formulation for 3D Magnetostatics in the Space H(rot)∩ H(div) February 1997 97-2 Joachim Schöberl Robust Multigrid Preconditioning for Parameter Dependent Problems I: The Stokes-type Case. June 1997 97-3 Ferdinand Kickinger, Sergei V. Nepomnyaschikh, Ralf Pfau, Joachim Schöberl Numerical Estimates of Inequalities in H 1 2 . August 1997 97-4 Joachim Schöberl Programmbeschreibung NAOMI 2D und Algebraic Multigrid. September 1997 From 1998 to 2008 technical reports were published by SFB013. Please see http://www.sfb013.uni-linz.ac.at/index.php?id=reports From 2004 on reports were also published by RICAM. Please see http://www.ricam.oeaw.ac.at/publications/list/ For a complete list of NuMa reports see http://www.numa.uni-linz.ac.at/Publications/List/ ABSTRACT ROBUST COARSE SPACES FOR SYSTEMS OF PDES VIA GENERALIZED EIGENPROBLEMS IN THE OVERLAPSROBUST COARSE SPACES FOR SYSTEMS OF PDES VIA GENERALIZED EIGENPROBLEMS IN THE OVERLAPS NICOLE SPILLANE, VICTORITA DOLEAN, PATRICE HAURET, FRÉDÉRIK NATAF, CLEMENS PECHSTEIN, AND ROBERT SCHEICHL Abstract. Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property. Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property.