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High order FEM, fast solvers for tensor product elements

Tuesday, February 15, 3:30PM
POB 6.304

Sven Beuchler

The talk deals with the discretization of elliptic problems by the hp-version of the finite element method. In the first part, several polynomial bases which yield in a sparse system matrix are presented. The main part of this talk is devoted to the efficient solution of the system of linear algebraic equations. From the literature it is known that the conjugate gradient method with domain decomposition preconditioners is one of the most efficient methods for solving systems of linear algebraic equations resulting from hp-version. The ingredients of a nonoverlapping preconditioner are a preconditioner for the Schur complement, a preconditioner related to the Dirichlet problems in the subdomains, and an extension operator from the boundaries of the subdomains into their interior. Thi yields to quasioptimal solvers in two space dimensions. For overlapping preconditioners, a preconditioner related to the Dirichlet problems in the subdomains and coarse space component is required. It has been proved by Pavarino that overlapping partition of the domain into patches yields to an optimal preconditioner in hp-FEM in two and three space dimensions. For the solution of the subproblems we intend to use the tensor product strcuture of the patches combined with wavelet methods. The theoretical results are confirmed by several numerical examples. In the last part of the talk, we consider the design of hp-finite element methods for optimal control problems.

The results are obtained in collaboration with C. Schwab, J. Schoeberl,V. Pillwein, S. Zaglmayr, T. Eibner, R. Schneider, D. Braess, U. Langer, C. Pechstein and D. Wachsmuth.

Host: L. Demkowicz