Efficient Preconditioned Solution Methods for Elliptic Partial Differential Equations, pp.i-ii
The present book is a collection of chapters written by outstanding specialists, and is devoted to one of the most challenging topics of contemporary numerical mathematics. The topic ? development of efficient preconditioners and solution methods for (discretized) elliptic partial differential equations ? is crucial for the mathematical modelling of complex physical and other processes in most branches of science and engineering. Efficient solution methods usually arise as a combination of a suitable iterative technique and an efficient preconditioner, which is typically problem specific, using a proper approximation to the solved problem with further attributes as cheap actions with it, a possibility of parallel implementation, numerical and parallel scalability and robustness with respect to the problem parameters. The development of preconditioners has now its own history starting from matrix splittings, diagonal scaling and incomplete factorization for model elliptic problems and continuing with multigrid and multilevel methods, approximate inverses, domain decomposition methods and other approaches to the solution of complicated elliptic problems discretized by the finite element or other discretization methods. A substantial progress in the field of preconditioners has been achieved, but at the same time many new questions and challenges have arisen, and this is why the research is even more intensive nowadays than it was previously. The present book covers many topics of the contemporary research in the field of efficient preconditioners, namely: Chapter 1, written by P. Boyanova and S. Margenov, concerns the Algebraic Multilevel Iterative (AMLI) methods first introduced by Axelsson and Vassilevski. The paper overviews the methods with a special care of efficiency and robustness as well as application to both elliptic and parabolic problems discretized by either conforming or nonconforming finite element methods......
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