Dirichlet-dirichlet 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.
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2 Fundamentals of the Schwarz Methods
3 Overlapping Domain Decomposition Methods
4 Nonoverlapping DD Methods for h FE Discretizations in 2d
5 BPStype DD Preconditioners for 3d Elliptic Problems
6 DD Algorithms for Discretizations with Chaotically Piecewise Variable Orthotropism
7 Nonoverlapping DD Methods for hp Discretizations of 2d Elliptic Equations
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basis bilinear form block diagonal bound Cauchy inequality coarse coefficients components computational coordinate polynomials corresponding DD algorithms DD methods DD preconditioner DD solver DDloc decomposition mesh defined denoted diag Dirichlet boundary condition Dirichlet problems Dirichlet–Dirichlet domain decomposition Domain Decomposition Methods edge efficient elliptic problems estimate faces fast solvers FE functions FE space finite element finite element method finite-difference grid hierarchical reference element hp discretizations inequalities inexact inter-subdomain interface introduce Korneev Lemma linear multigrid multigrid method multilevel nodes norm notations optimal orthogonal overlapping piecewise positive constants preconditioning prolongation operator proof quadratic form relative condition number respectively satisfying Schur complement preconditioner seminorm shape regularity Sobolev spaces solving spectral reference elements spectrally equivalent stiffness matrix subdomains subdomains of decomposition Subsection subspace superelements Theorem triangulation vector vertex wavelet wire basket