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Hybrid and Multiplicative Overlapping Schwarz Algorithms with Standard Coarse Spaces for Mixed Linear Elasticity and Stokes Problems

Part of: Numerical linear algebra Partial differential equations, boundary value problems Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  05 October 2016

Mingchao Cai  and
Luca F. Pavarino
Mingchao Cai*
Affiliation:
Department of Mathematics, Morgan State University, 1700 E. Cold Spring Ln, Baltimore, MD 21251, USA
Luca F. Pavarino*
Affiliation:
Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133 Milano, Italy
*
*Corresponding author. Email addresses:cmchao2005@gmail.com (M. Cai), luca.pavarino@unimi.it (L. F. Pavarino)
*Corresponding author. Email addresses:cmchao2005@gmail.com (M. Cai), luca.pavarino@unimi.it (L. F. Pavarino)
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Abstract

The goal of this work is to construct and study hybrid and multiplicative two-level overlapping Schwarz algorithms with standard coarse spaces for the almost incompressible linear elasticity and Stokes systems, discretized by mixed finite and spectral element methods with discontinuous pressures. Two different approaches are considered to solve the resulting saddle point systems: a) a preconditioned conjugate gradient (PCG) method applied to the symmetric positive definite reformulation of the almost incompressible linear elasticity system obtained by eliminating the pressure unknowns; b) a GMRES method with indefinite overlapping Schwarz preconditioner applied directly to the saddle point formulation of both the elasticity and Stokes systems. Condition number estimates and convergence properties of the proposed hybrid and multiplicative overlapping Schwarz algorithms are proven for the positive definite reformulation of almost incompressible elasticity. These results are based on our previous study [8] where only additive Schwarz preconditioners were considered for almost incompressible elasticity. Extensive numerical experiments with both finite and spectral elements show that the proposed overlapping Schwarz preconditioners are scalable, quasi-optimal in the number of unknowns across individual subdomains and robust with respect to discontinuities of the material parameters across subdomains interfaces. The results indicate that the proposed preconditioners retain a good performance also when the quasi-monotonicity assumption, required by the available theory, does not hold.

Keywords

Overlapping Schwarz preconditionersalmost incompressible linear elasticityStokes equationssaddle point problemsfinite and spectral elements

MSC classification

Secondary: 65M55: Multigrid methods; domain decomposition 65N22: Solution of discretized equations 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65F08: Preconditioners for iterative methods 65M22: Solution of discretized equations
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 4 , October 2016 , pp. 989 - 1015
Copyright
Copyright © Global-Science Press 2016 

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