Balancing Neumann-Neumann Preconditioners for the Mixed Formulation of Almost-Incompressible Linear Elasticity

Balancing Neumann-Neumann methods are extended to the equations arising from the mixed formulation of almost-incompressible linear elasticity problems discretized with discontinuous-pressure finite elements. This family of domain decomposition algorithms has previously been shown to be effective for large finite element approximations of positive definite elliptic problems. Our methods are proved to be scalable and to depend weakly on the size of the local problems. Our work is an extension of previous work by Pavarino and Widlund on BNN methods for Stokes equation. Our iterative substructuring methods are based on the partition of the unknowns into interior ones — including interior displacements and pressures with zero average on every subdomain — and interface ones — displacements on the geometric interface and constant-by-subdomain pressures. The restriction of the problem to the interior degrees of freedom is then a collection of decoupled local problems that are well-posed even in the incompressible limit. The interior variables are eliminated and a hybrid preconditioner of BNN type is designed for the Schur complement problem. The iterates are restricted to a benign subspace, on which the preconditioned operator is positive definite, allowing for the use of conjugate gradient methods. A complete convergence analysis of the method is presented for the constant coefficient case. The algorithm is extended to handle discontinuous coefficients, but a full analysis is not provided. Extensions of the algorithm and of the analysis are also presented for problems combining pure-displacement and mixed finite elements in different subregions. An algorithm is also proposed for problems with continuous discrete pressure spaces. All the algorithms discussed have been implemented in parallel codes that have been successfully tested on large sample problems on large parallel computers; results are presented and discussed. Implementations issues are also discussed, including a version of our main algorithm that does not require the solution of any auxiliary saddle-point problem since all subproblems of the

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