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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Fundamentals of the Schwarz Methods
Overlapping Domain Decomposition Methods
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according allows application arithmetical assume basis block bound boundary coarse coefficients complete components computational condition number consequence consider constants coordinate corresponding cost DD algorithms DD preconditioner decomposition defined definition denoted depending diag diagonal direct Dirichlet problems discretizations domain decomposition edge efficient elliptic equations estimate faces finite element follows functions given grid hierarchical hold inequalities interface internal introduce iterative Korneev Lemma linear means mesh methods multiplications nodes norm notations obtained operator optimal polynomials positive preconditioner preconditioner-solver preconditioning problems procedure proof properties proved quasiuniformity reference element relative represented requires respectively satisfying Schur complement Schur complement preconditioner shape similar solution solvers solving space spectrally equivalent square step stiffness matrix subdomains Subsection subspace sufficiently term Theorem traces transformed triangulation values vector vertex vertices wire basket