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Finite and boundary element tearing and interconnecting solvers for multiscale problems. (English) Zbl 1272.65100

Lecture Notes in Computational Science and Engineering 90. Berlin: Springer (ISBN 978-3-642-23587-0/hbk; 978-3-642-23588-7/ebook). xiv, 312 p. (2013).
The book gives a detailed and self-contained presentation of tearing and interconnecting methods for finite and boundary element discretizations of second-order elliptic partial differential equations. These methods belong to the class of non-overlapping domain decomposition solvers and are used as coupling methods between possibly different discretizations and iterative solvers for large-scale equations, which are well-suited for parallelization. The presentation includes a description of the corresponding algorithms and a rigorous convergence theory together with the results of some numerical tests. Besides finite element tearing and interconnecting methods (FETI), the dual-primal FETI and balancing domain decomposition by constrains (BDDC), already addressed in previous monographs by A. Toselli and O. Widlund [Domain decomposition methods – algorithms and theory. Springer Series in Computational Mathematics 34. Berlin: Springer (2005; Zbl 1069.65138)] and T. P. A. Mathew [Domain decomposition methods for the numerical solution of partial differential equations. Lecture Notes in Computational Science and Engineering 61. Berlin: Springer (2008; Zbl 1147.65101)], the book treats boundary element tearing and interconnecting (BETI), the coupling FETI/BETI, and the case of highly varying coefficients, not resolved by the subdomain partitioning.
The basic idea of FETI/BETI methods is to subdivide the computational domain into smaller subdomains, where the corresponding local problems can still be handled efficiently by direct solvers. The coupling of the local problems is performed by Lagrangian multipliers. The global solution is then obtained from this dual problem which is usually solved iteratively by the repeated solution of local problems. Here, suitable preconditioners are needed in order to ensure that the number of iterations depends only weakly on the size of the local problems.
The book has 5 chapters: 1. Preliminaries; 2. One-level FETI/BETI methods; 3. Multiscale problems; 4. Unbounded domains; 5. Dual-primal methods.
Chapter 1 collects standard results needed for a self-contained representation of the material. The subject of Chapter 2 is the coupling of finite and boundary element discretizations within the tearing and interconnecting framework for scalar second-order elliptic equations in a bounded domain, where the diffusion coefficient is constant on each subdomain. The author discusses different formulations of FETI/BETI methods, analyses unpreconditioned methods (which turn out to be sub-optimal) and provides a rigorous analysis of the preconditioned method with the so-called scaled Dirichlet preconditioner. It is proved that the condition number of the corresponding preconditioned system is bounded in terms of a logarithmic expression in the local problem size. Furthermore, the bound is independent of jumps in the diffusion coefficient across subdomain interfaces.
Chapter 3 treats FETI methods for diffusion equations with highly heterogeneous coefficient distributions, a rather new and growing research area. Using new technical tools such as weighted Poincaré inequalities rigorous bounds for the condition number of the preconditioned FETI system are proved, that depend only on the coefficient variation in the vicinity of the subdomain interfaces. It is shown, that if the coefficient varies only moderately in a layer near the boundary of each subdomain, then the method is proved to be robust with respect to arbitrary variation in the interior of each subdomain and with respect to coefficient jumps across subdomain interfaces.
In Chapter 4, the methods of Chapter 2 are extended to the case, that one subdomain corresponds to an exterior unbounded problem, while the other subdomains are bounded. The exterior problem is approximated using the boundary element method. The fact that this domain can touch arbitrarily many interior subdomains and that the diameter of its boundary is larger than those of the other subdomains leads to special difficulties. The improved analysis delivers two types of explicit condition number bounds that depend on a few geometric parameters, and which are quasi-optimal in special cases.
In Chapter 5, dual-primal (DP) FETI/BETI methods and BDDC methods are considered. Here unknowns at subdomain corners are kept primal and possibly the equality of averages across edges or faces on subdomain interfaces is enforced. After the elimination of the primal unknowns in many cases the resulting subdomain problems are easier to handle. The extension of the DP methods to the unbounded case and to some problems from Chapter 3 is discussed.
This self-contained monograph essentially contributes to the theory of the finite and boundary element solution of multiscale problems by tearing and interconnecting methods. It is a good complement to existing monographs and surveys about this active research field. The text can serve also as a useful reference book and is intended for researchers, postgraduate students, and all practitioners working in the areas of efficient solution methods for partial differential equations.

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F08 Preconditioners for iterative methods
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N38 Boundary element methods for boundary value problems involving PDEs
65Y05 Parallel numerical computation
35J25 Boundary value problems for second-order elliptic equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis

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