Seminar in Numerical Analysis: Marco Zank (U Wien)

Conforming Space-Time Finite Element Methods

For the discretization of time-dependent partial differential equations, the standard approaches are explicit or implicit time-stepping schemes together with finite element methods in space. An alternative approach is the usage of space-time methods, where the space-time domain is discretized and the resulting global linear system is solved at once. In this talk, some recent developments in space-time finite element methods are reviewed. For this purpose, the heat equation and the wave equation serve as model problems. First, for both model problems, space-time variational formulations and their unique solvability in space-time Sobolev spaces are discussed, where a modified Hilbert transformation is used such that ansatz and test spaces are equal. Second, conforming discretization schemes, using piecewise polynomial, globally continuous functions, are introduced. Solvability and stability of these numerical schemes are discussed. Next, we investigate efficient direct solvers for the occurring huge linear systems. The developed solvers are based on the Bartels--Stewart method and on the Fast Diagonalization method, which result in solving a sequence of spatial subproblems. The solver based on the Fast Diagonalization method allows solving these spatial subproblems in parallel, leading to a full parallelization in time. In the last part of the talk, numerical examples are shown and discussed.

For further information about the seminar, please visit this webpage.

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