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.
Fundamentals of the Schwarz Methods
Overlapping Domain Decomposition Methods
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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 decomposition mesh defined denoted diag diagonal matrix Dirichlet boundary condition Dirichlet problems domain decomposition Domain Decomposition Methods edge efficient elliptic equations estimate faces fast solvers FE functions FE space finite element finite element method finite-difference grid h-version H¹(N hierarchical reference element hp discretizations inequalities inexact inter-subdomain interface introduce Korneev Lemma linear matrix-vector multiplications multigrid multigrid method multilevel nodes norm notations optimal orthogonal p-elements piecewise positive constants preconditioning procedure prolongation operator proof quadratic form reference element relative condition number respectively satisfying Schur complement preconditioner seminorm shape regularity solving spectral reference elements spectrally equivalent stiffness matrix subdomains subdomains of decomposition Subsection subspace superelements Theorem triangulation V₁ vector vertex vertices wavelet Widlund wire basket