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

Abstract

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.

On the Design of Small Coarse Spaces for Domain Decomposition Algorithms

“…If the coarse basis functions originate from (1) rather than (2,5), then it follows from elementary estimates and Lemma 3 that 1 + log(H i /h i ) will appear as an additional factor on the right-hand-side of (6). Thus, this additional factor will also be present in (14) and (15).…”

“…In our proof, we will rely on the fact that the sum of the basis functions of the coarse nodes of any subdomain will give us the basis functions for the space of rigid body modes. This follows from the fact that we have a partition of unity defined by the p nc ; see (1) or (2) and (5).…”

“…Results similar to those of [12] have recently been developed for almost incompressible elasticity in [4,5] where the coarse space is a conventional one based on a coarse triangulation of Ω; the Lamé parameter µ is essentially assumed to meet the requirements of Assumption 1 of this paper. In contrast to the algorithm of those studies, the coarse spaces of this study are also well defined for quite arbitrary subdomains generated, e.g., by a mesh partitioner.…”

“…, 4, be the distance between a node n ∈ N on the face and c i , the ith ancestor coarse node. We then assign the values (5)…”

“…(6). In our experiments with Option 2, we will use formula (2) for all cases when there are three or fewer ancestor nodes and formula (5) otherwise.…”