-A DD (domain decomposition) preconditioner of almost optimal in p arithmetical complexity is presented for the hierarchical hp discretizations of 3-d second order elliptic equations. We adapt the wire basket substructuring technique to the hierarchical hp discretization, obtain a fast preconditioner-solver for faces by Kinterpolation technique and show that a secondary iterative process may be efficiently used for prolongations from faces. The fast solver for local Dirichlet problems on subdomains of decomposition is based on our earlier derived finite-difference like preconditioner for the internal stiffness matrices of p-finite elements and fast solution procedures for systems with this preconditioner, which appeared recently. The relative condition number, provided by the DD preconditioner under consideration, is O ((1 + log p) 3.5 ) and its total arithmetic cost is O ((1+log p)

Domain decomposition methods nowadays provide powerful tools for constructing efficient parallel solvers for large‐scale systems of algebraic equations arising from the discretization of partial differential equations. The classical alternating Schwarz method and the classical substructuring technique have led to advanced overlapping and nonoverlapping domain decomposition solvers (preconditioners), that can be analyzed from a unified point of view now called Schwarz theory . This survey article starts with a brief historical overview, provides the basic results of the Schwarz theory, looks at some overlapping domain decomposition methods (preconditioners) in brief, and discusses more extensively various nonoverlapping domain decomposition techniques.

PrefaceDomain 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. The Alternating Schwarz Method, proposed by H. Schwarz (1869) in order to prove the existence of harmonic functions with prescribed Dirichlet data on the boundary of complicated domains, and substructuring techniques, developed by engineers in the 60s of the preceding century in order to create faster procedures for the analysis of complex structures, are commonly accepted as the origins of modern DD methods. Thus, DD methods have been developed for a long time, but most extensively since the first international DD conference that was held at Paris in 1987. This concerns both the theory and the practical use of DD techniques for creating efficient application software for massive parallel computers. The advances in DD are well documented in the proceedings of the international DD conferences 1 since 1987 and numerous papers. The first textbook on DD methods was published by B.F. Smith, P.E. Bjørstad, and W. Dirichlet-Dirichlet Domain Decomposition Methods for Elliptic ProblemsPechstein has recently published a monograph on Finite Element Tearing and Interconnecting (FETI) and Boundary Element Tearing and Interconnecting (BETI) methods for a special class of multiscale problems. Our book is different from the textbooks mentioned above and very different from Pechstein's monograph. We mainly discuss one special class of primal substructuring methods also called Dirichlet-Dirichlet DD methods, and we emphasize the peculiarities of their application to hp finite element equations. In particular, we discuss and analyze the inexact versions of these DD methods which lead to optimal or, at least, almost optimal complexity and high efficiency in practical applications. The optimization of several important components of DD algorithms for hp discretization was achieved only quite recently. These topics are not discussed or at least not sufficiently discussed in the books mentioned above. The contributions of the authors to this field mainly appeared in journal publications including joint papers. The reader will become familiar with inexact Dirichlet-Dirichlet DD methods enjoying optimal complexity and the techniques for the numerical analysis of these methods. Thus, we hope that readers interested in both practice and theory can benefit from our book. In particular, the book will open the possibility for the reader to use such DD methods in new fields of applications like multiphysics applications in Computational Sciences. In this book, the reader can find a brief historical overview, the basic results of the general theory of domain and space decompositions as well as the description and analysis of practical DD algorithms. Elliptic problems with strongly jumping coefficients are daily met in the engineering practice. Numerical techniques should not deteriorate in efficiency when solvin...

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