A class of parallel incomplete factorization preconditionings for the solution of large linear systems is investigated. The approach may be regarded as a generalized domain decomposition method. Adjacent subdomains have to communicate during the setting up of the precon-ditioner, and during the application of the preconditioner. Overlap is not necessary to achieve high performance. Fill-in leve...

Rigorous computer simulations of propagating electromagnetic fields have become an important tool for optical metrology and design of nanostructured optical components. A vectorial finite element method (FEM) is a good choice for an accurate modeling of complicated geometrical features. However, from a numerical point of view solving the arising system of linear equations is very demanding even...

Several a posteriori error estimators are introduced and analyzed for a discontinuous Galerkin formulation of a model second-order elliptic problem. In addition to residual-type estimators, we introduce some estimators that are couched in the ideas and techniques of domain decomposition. Results of numerical experiments are presented.

A preconditioner for iterative solution of the interface problem in Schur Complement Domain Decomposition Methods is presented. This preconditioner is based on solving a problem in a narrow strip around the interface. It requires much less memory and computing time than classical Neumann-Neumann preconditioner and its variants, and handles correctly the flux splitting among subdomains that shar...

Domain decomposition methods without overlapping for the approximation of parabolic problems are considered. Two kinds of methods are discussed. In the rst method systems of algebraic equations resulting from the approximation on each time level are solved iteratively with a Neumann-Dirichlet preconditioner. The second method is direct and similar to certain iterative methods with a Neumann-Neu...

The purpose of this paper is to bring to the domain decomposition community certain implications of a new operator trigonometry and of the Robin boundary condition as they pertain to domain decomposition methods and theory. In Section 2 we recall some basic facts and recent results concerning the new operator trigonometry as it applies to iterative methods. This theory reveals that the converge...

This paper presents a new non-overlapping domain decomposition method for the Helmholtz equation, whose effective convergence is quasi-optimal. These improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the Dirichlet to Neumann operator. A convergence theorem of the algorithm is established and numerical results valida...

More than a decade ago, Bramble, Pasciak and Xu developed a framework in analyzing the multigrid methods with nonnested spaces or noninherited quadratic forms. It was subsequently known as the BPX multigrid framework, which was widely used in the analysis of multigrid and domain decomposition methods. However, the framework has an apparent limit in the analysis of nonnested V-cycle methods, and...

More than a decade ago, Bramble, Pasciak and Xu developed a framework in analyzing the multigrid methods with nonnested spaces or noninherited quadratic forms. It was subsequently known as the BPX multigrid framework, which was widely used in the analysis of multigrid and domain decomposition methods. However, the framework has an apparent limit in the analysis of nonnested V-cycle methods, and...