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Iterative solvers

Efficient iterative solution techniques for linear systems of equations arising from the discretization of boundary value problems are of special interest.Very often huge systems are obtained, with condition numbers which depend on the meshsize h of the triangulation, which typically grow in proportion to h-2. Then, classical iteration schemes like Jacobi- or SOR-type methods result in very slow convergence rates. Nonlinear iteration schemes as the conjugate gradient method give better results with a convergence rate depending on the square root of the condition number. Schwarz methods provide a powerful tool for the efficient iterative solution of the huge systems of equations. Multigrid methods which are optimal or preconditioned conjugate gradient methods are of special interest. Very often the preconditioner is built from the solution of subproblems of less complexity which are either related to a decomposition of the geometrical domain into subdomains (iterative substructuring, overlapping dd, Dirichlet-Neumann) or a hierarchical splitting of the finite element space into subspaces (multilevel).

Introduction to solvers (PDF in German)