DirichletDirichlet Domain Decomposition Methods for Elliptic Problems

h and hp Finite Element Discretizations

Domain decomposition (DD) methods provide powerful tools for constructing parallel numerical solution algorithms for large scale systems of algebraic equations arising from the discretization of partial differential equations. These methods are well-established and belong to a fast developing area. In this volume, the reader will find a brief historical overview, the basic results of the general theory of domain and space decomposition methods as well as the description and analysis of practical DD algorithms for parallel computing. It is typical to find in this volume that most of the presented DD solvers belong to the family of fast algorithms, where each component is efficient with respect to the arithmetical work. Readers will discover new analysis results for both the well-known basic DD solvers and some DD methods recently devised by the authors, e.g., for elliptic problems with varying chaotically piecewise constant orthotropism without restrictions on the finite aspect ratios.

The hp finite element discretizations, in particular, by spectral elements of elliptic equations are given significant attention in current research and applications. This volume is the first to feature all components of Dirichlet–Dirichlet-type DD solvers for hp discretizations devised as numerical procedures which result in DD solvers that are almost optimal with respect to the computational work. The most important DD solvers are presented in the matrix/vector form algorithms that are convenient for practical use.

Contents:

• Introduction
• Fundamentals of the Schwarz Methods
• Overlapping Domain Decomposition Methods
• Nonoverlapping DD Methods for h FE Discretizations in 2d
• BPS-type DD Preconditioners for 3d Elliptic Problems
• DD Algorithms for Discretizations with Chaotically Piecewise Variable Orthotropism
• Nonoverlapping DD Methods for hp Discretizations of 2d Elliptic Equations
• Fast Dirichlet Solvers for 2d Reference Elements
• Nonoverlapping Dirichlet–Dirichlet Methods for hp Discretizations of 3d Elliptic Equations

Readership: Graduate students and researchers in academia, laboratories and research institutions, and industry.
Key Features:

• Optimal in computational work and robust domain decomposition (DD) solvers for h- and hp-finite element equations
• Unique treatment of DD algorithms for hp spectral discretizations
• Includes analysis and algorithmic descriptions of DD methods
• Written by famous experts in finite element methods and numerical analysis