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Domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations of partial differential equations. The preconditioners are constructed from exact or approximate solvers for the same partial differential equation restricted to a set of subregions into which the given region has been divided. In addition, the preconditioner is often augmented by a coarse, second-level approximation that provides additional, global exchange of information that can enhance the rate of convergence considerably. The iterative substructuring methods, based on decompositions of the region into nonoverlapping subregions, form one of the main families of such algorithms.Many domain decomposition algorithms can conveniently be described and analyzed as Schwarz methods. These algorithms are fully defined in terms of a set of subspaces and auxiliary bilinear forms.A general theoretical framework has previously been developed. In this paper, these techniques are used in an analysis of iterative substructuring methods for elliptic problems in three dimensions. A special emphasis is placed on the difficult problem of designing good coarse models and obtaining robust methods for which the rate of convergence is insensitive to large variations in the coefficients of the differential equation.Domain decomposition algorithms can conveniently be built from modules that represent local and global components of the preconditioner. In this paper, a number of such possibilities are explored, and it is demonstrated how a great variety of fast algorithms can be designed and analyzed.

Abstract. Most efficient linear solvers use composable algorithmic components, with the most common model being the combination of a Krylov accelerator and one or more preconditioners.A similar set of concepts may be used for nonlinear algebraic systems, where nonlinear composition of different nonlinear solvers may significantly improve the time to solution. We describe the basic concepts of nonlinear composition and preconditioning and present a number of solvers applicable to nonlinear partial differential equations. We have developed a software framework in order to easily explore the possible combinations of solvers. We show that the performance gains from using composed solvers can be substantial compared with gains from standard Newton-Krylov methods.

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We propose a robust interpolation for multigrid based on the concepts of energy minimization and approximation. The formulation is general; it can be applied to any dimensions. The analysis for one dimension proves that the convergence rate of the resulting multigrid method is independent of the coe cient of the underlying PDE, in addition to being independent of the mesh size. We demonstrate numerically the e ectiveness of the multigrid method in two dimensions by applying it to a discontinuous coe cient problem and an oscillatory coe cient problem. We also show using a one-dimensional Helmholtz problem that the energy minimization principle can be applied to solving elliptic problems that are not positive de nite. 1. Introduction. Multigrid methods are widely used as e cient solvers for second order elliptic partial di erential equations (PDEs) because of their often optimal convergence behavior; that is, their convergence rate is independent of the mesh size. Optimal theory can be found, for example, in 2, 3, 4, 20, 27, 31, 38, 39]. However, the convergence rate may depend on the nature of the coe cients in the PDE. Typically, the convergence deteriorates as the coe cients become rougher. Speci cally, if the coe cients are anisotropic 20], have large jumps 1, 5, 10, 11] or are highly oscillatory 17, 26, 34], standard multigrid methods will converge very slowly. Special techniques such as line Gauss-Seidel/block smoothing 5], semi-coarsening 12, 13, 32], algebraic multigrid 6, 28, 30, 33], frequency decomposition 14, 21, 34], and homogenization 17, 26] are used to handle some of these cases. In this paper, we study the design of multigrid methods from the energy minimization point of view, which gives powerful insight into the design of robust multigrid methods. The success of multigrid hinges on the choice of the coarse grid points, the smoothing procedure, the interpolation operators, and the coarser grid discretization. In standard multigrid, full coarsening, Jacobi or Gauss-Seidel smoothing, and linear interpolation are usually used. Classical convergence theory shows that these simple ingredients are enough to achieve optimal convergence for smooth coe cient problems. In general, however, these choices may lead to slow convergence. In one dimension, to remedy the situation, a more robust interpolation 20, 28, 36] can be used. It is obtained by solving local homogeneous PDEs, which are equivalent to minimizing the energy of the coarse grid basis functions. The extension of this approach to higher dimensions is not obvious. Nonetheless, many attempts 1, 10, 20, 19, 24, 29, 36] have been made to set up similar local PDEs for de ning a robust interpolation. In place of setting up PDEs, we consider an equivalent minimization formulation and derive a so-called energy-minimizing interpolation with special emphasis on its stability and approximation properties, which are essential for optimal convergence. This approach to determining appropriate interpolation operators has also been used for iterative substructuri...