Hybrid and Multiplicative Overlapping Schwarz Algorithms with Standard Coarse Spaces for Mixed Linear Elasticity and Stokes Problems (Communications in Computational Physics, 2018)
The goal of this work is to construct and study hybrid and multiplicativetwo-level overlapping Schwarz algorithms with standard coarse spaces for the almostincompressible linear elasticity and Stokes systems, discretized by mixed finite andspectral element methods with discontinuous pressures. Two different approaches areconsidered to solve the resulting saddle point systems: a) a preconditioned conjugategradient (PCG) method applied to the symmetric positive definite reformulation ofthe almost incompressible linear elasticity system obtained by eliminating the pressureunknowns; b) a GMRES method with indefinite overlapping Schwarz preconditionerapplied directly to the saddle point formulation of both the elasticity and Stokessystems. Condition number estimates and convergence properties of the proposed hybridand multiplicative overlapping Schwarz algorithms are proven for the positivedefinite reformulation of almost incompressible elasticity. These results are based onour previous study [8] where only additive Schwarz preconditioners were consideredfor almost incompressible elasticity. Extensive numerical experiments with both finiteand spectral elements show that the proposed overlapping Schwarz preconditionersare scalable, quasi-optimal in the number of unknowns across individual subdomainsand robust with respect to discontinuities of the material parameters across subdomainsinterfaces. The results indicate that the proposed preconditioners retain a goodperformance also when the quasi-monotonicity assumption, required by the availabletheory, does not hold.