Publications

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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 Balancing Domain Decomposition by Constraints (BDDC) method is the most advanced method from the Balancing family of iterative substructuring methods for the solution of large systems of linear algebraic equations arising from discretization of elliptic boundary value problems. In the case of many substructures, solving the coarse problem exactly becomes a bottleneck. Since the coarse problem in BDDC has the same structure as the original problem, it is straightforward to apply the BDDC method recursively to solve the coarse problem only approximately. In this paper, we formulate a new family of abstract Multispace BDDC methods and give condition number bounds from the abstract additive Schwarz preconditioning theory. The Multilevel BDDC is then treated as a special case of the Multispace BDDC and abstract multilevel condition number bounds are given. The abstract bounds yield polylogarithmic condition number bounds for an arbitrary fixed number of levels and scalar elliptic problems discretized by finite elements in two and three spatial dimensions. Numerical experiments confirm the theory. © 2008 Springer-Verlag.

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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.

We present a new family of stabilized methods for the Stokes problem. The focus of the paper is on the lowest order velocity-pressure pairs. While not LBB compliant, their simplicity and attractive computational properties make these pairs a popular choice in engineering practice. Our stabilization approach is motivated by terms that characterize the LBB "deficiency" of the unstable spaces. The stabilized methods are defined by using these terms to modify the saddle-point Lagrangian associated with the Stokes equations. The new stabilized methods offer a number of attractive computational properties. In contrast to other stabilization procedures, they are parameter free, do not require calculation of higher order derivatives or edge-based data structures, and always lead to symmetric linear systems. Furthermore, the new methods are unconditionally stable, achieve optimal accuracy with respect to solution regularity, and have simple and straightforward implementations. We present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods. © 2006 Society for Industrial and Applied Mathematics.

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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.

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