In this talk, we present a collection of domain decomposition algorithms for mixed finite element formulations of elasticity and incompressible fluids. The key component of each of these algorithms is the coarse space. Here, the coarse spaces are obtained in an algebraic manner by harmonically extending coarse boundary data. Various aspects of the coarse spaces are discussed for both continuous and discontinuous interpolation of pressure. Further, both classical overlapping Schwarz and hybrid iterative substructuring preconditioners are described. Numerical results are presented for almost incompressible elasticity and the Navier Stokes equations which demonstrate the utility of the methods for both structured and irregular mesh decompositions. We also discuss a simple residual scaling approach which often leads to significant reductions in iterations for these algorithms.
Advanced computing hardware and software written to exploit massively parallel architectures greatly facilitate the computation of extremely large problems. On the other hand, these tools, though enabling higher fidelity models, have often resulted in much longer run-times and turn-around-times in providing answers to engineering problems. The impediments include smaller elements and consequently smaller time steps, much larger systems of equations to solve, and the inclusion of nonlinearities that had been ignored in days when lower fidelity models were the norm. The research effort reported focuses on the accelerating the analysis process for structural dynamics though combinations of model reduction and mitigation of some factors that lead to over-meshing.
A simple and effective approach is presented to construct coarse spaces for overlapping Schwarz preconditioners. The approach is based on energy minimizing extensions of coarse trace spaces, and can be viewed as a generalization of earlier work by Dryja, Smith, and Widlund. The use of these coarse spaces in overlapping Schwarz preconditioners leads to condition numbers bounded by C(1 + H/δ)(1 + log(H/h)) for certain problems when coefficient jumps are aligned with subdomain boundaries. For problems without coefficient jumps, it is possible to remove the log(H/h) factor in this bound by a suitable enrichment of the coarse space. Comparisons are made with the coarse spaces of two other substructuring preconditioners. Numerical examples are also presented for a variety of problems.
The focus of this paper is a penalty-based strategy for preconditioning elliptic saddle point systems. As the starting point, we consider the regularization approach of Axelsson in which a related linear system, differing only in the (2,2) block of the coefficient matrix, is introduced. By choosing this block to be negative definite, the dual unknowns of the related system can be eliminated resulting in a positive definite primal Schur complement. Rather than solving the Schur complement system exactly, an approximate solution is obtained using a substructuring preconditioner. The approximate primal solution together with the recovered dual solution then define the preconditioned residual for the original system.
The focus of this paper is a penalty-based strategy for preconditioning elliptic saddle point systems. As the starting point, we consider the regularization approach of Axelsson in which a related linear system, differing only in the (2,2) block of the coefficient matrix, is introduced. By choosing this block to be negative definite, the dual unknowns of the related system can be eliminated resulting in a positive definite primal Schur complement. Rather than solving the Schur complement system exactly, an approximate solution is obtained using a substructuring preconditioner. The approximate primal solution together with the recovered dual solution then define the preconditioned residual for the original system. The approach can be applied to a variety of different saddle point problems. Although the preconditioner itself is symmetric and indefinite, all the eigenvalues of the preconditioned system are real and positive if certain conditions hold. Stronger conditions also ensure that the eigenvalues are bounded independently of mesh parameters. An interesting feature of the approach is that conjugate gradients can be used as the iterative solution method rather than GMRES. The effectiveness of the overall strategy hinges on the preconditioner for the primal Schur complement. Interestingly, the primary condition ensuring real and positive eigenvalues is satisfied automatically in certain instances if a Balancing Domain Decomposition by Constraints (BDDC) preconditioner is used. Following an overview of BDDC, we show how its constraints can be chosen to ensure insensitivity to parameter choices in the (2,2) block for problems with a divergence constraint. Examples for different saddle point problems are presented and comparisons made with other approaches.
