Olof B. Widlund

Iterative methods for the solution of elliptic problems on regions partitioned into substructures

Finite element problems can often naturally be divided into subproblems which correspond to subregions into which the region has been partitioned or from which it was originally assembled. A class of iterative methods is discussed in which these subproblems are solved by direct methods, while the interaction across the curves or surfaces which divide the region is handled by a conjugate gradient method. A mathematical framework for this work is provided by regularity theory for elliptic finite element problems and by block Gaussian elimination. A full development of the theory, which shows that certain of these methods are optimal, is given for Lagrangian finite element approximations of second order linear elliptic problems in the plane. Results from numerical experiments are also reported.

Dual-Primal FETI Methods for Three-Dimensional Elliptic Problems with Heterogeneous Coefficients

In this paper, certain iterative substructuring methods with Lagrange multipliers are considered for elliptic problems in three dimensions. The algorithms belong to the family of dual-primal finite element tearing and interconnecting (FETI) methods which recently have been introduced and analyzed successfully for elliptic problems in the plane. The family of algorithms for three dimensions is extended and a full analysis is provided for the new algorithms. Particular attention is paid to finding algorithms with a small primal subspace since that subspace represents the only global part of the dual-primal preconditioner. It is shown that the condition numbers of several of the dual-primal FETI methods can be bounded polylogarithmically as a function of the dimension of the individual subregion problems and that the bounds are otherwise independent of the number of subdomains, the mesh size, and jumps in the coefficients. These results closely parallel those of other successful iterative substructuring methods of primal as well as dual type.

Schwarz Analysis of Iterative Substructuring Algorithms for Elliptic Problems in Three Dimensions

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...

FETI and Neumann--Neumann Iterative Substructuring Methods: Connections and New Results

The FETI and Neumann-Neumann families of algorithms are among the best know and most severely tested domain decomposition methods for elliptic partial differential equations. They are iterative substructuring methods and have many algorithmic components in common but there are also differences. The purpose of this paper is to further unify the theory for these two families of methods and to introduce a new family of FETI algorithms. Bounds on the rate of convergence, which are uniform with respect to the coefficients of a family of elliptic problems with heterogeneous coefficients, are established for these new algorithms. The theory for a variant of the Neumann--Neumann algorithm is also redeveloped stressing similarities to that for the FETI methods.

On the numerical solution of Helmholtz’s equation by the capacitance matrix method

In recent years the usefulness of fast Laplace solvers has been extended to problems on arbitrary regions in the plane by the development of capacitance matrix methods. The solution of the Dirichlet and Neumann problems for Helmholtzs equation is considered. It is shown, that by an appropriate choice of the fast solver, the capacitance matrix can be generated quite inexpensively. An analogy between capacitance matrix methods and classical potential theory for the solution of Laplaces equation is explored. This analogy suggests a modification of the method in the Dirichlet case. This new formulation leads to well-conditioned capacitance matrix equations which can be solved quite efficiently by the conjugate gradient method. A highly accurate solution can, therefore, be obtained at an expense which grows no faster than that for a fast Laplace solver on a rectangle when the mesh size is decreased.

A Lanczos Method for a Class of Nonsymmetric Systems of Linear Equations

Let L be a real linear operator with a positive definite symmetric part M. In certain applications a number of problems of the form

Domain decomposition algorithms for indefinite elliptic problems

Mv = g

Domain decomposition algorithms with small overlap

can be solved with less human or computational effort than the original equation

BDDC ALGORITHMS FOR INCOMPRESSIBLE STOKES EQUATIONS

Lu = f

Iterative Substructuring Preconditioners for Mortar Element Methods in Two Dimensions

. An iterative Lanczos method, which requires no a priori information on the spectrum of the operators, is derived for such problems. The convergence of the method is established assuming only that