# Mathias Anselmann: Application of higher order Galerkin-collocation time discretizations to waves and the Navier-Stokes equations with outlook towards Cut-FEM

Time: Thu 2019-12-12 13.15 - 14.00

Location:

Lecturer: Mathias Anselmann, Helmut Schmidt University, Hamburg

Abstract:

The families of Galerkin–collocation time discretization schemes, which establish a direct connection between the Galerkin method and the classical collocation schemes, is presented. They show the perspective of achieving the accuracy of the former with reduced computational costs, provided by the latter, in terms of less complex linear algebraic systems. Higher regularity in time of the discrete solutions is also ensured; cf [1, 2]. The method is first applied to the hyperbolic wave equation, written as first-order in time system:

$$\partial_t u - v = 0 \,, \quad \partial_t v - \nabla \cdot (c \nabla u) = f$$

and approved in a highly dynamic, numerical example.

Afterwards it is extended to the incompressible Navier–Stokes equations:

$$\partial_t \boldsymbol{v} + (\boldsymbol{v}\cdot \nabla )\boldsymbol{v} - \nu \Delta \boldsymbol{v} + \nabla p = \boldsymbol{f} \,, \quad \nabla \cdot \boldsymbol{v} = 0\,,$$

where Nitsche’s method is employed to impose all types of boundary condi- tions in a weak form. Next a flexible cut-cell integration scheme is shown and first dynamic Cut-FEM simulations are presented. These are the building block for capturing problems of fluid-structure interaction in the future.

References:

[1] M. Anselmann, M. Bause, Comparative study of continuously differentiable Galerkin time discretizations for the wave equation, Proceedings in Applied Mathematics & Mechanics 19 (1) (Jun. 2019). doi:10.2002/pamm.201900144

[2] M. Anselmann, M. Bause, S. Becher, G. Matthies, Galerkin-collocation approximation in time for the wave equation and its post-processing (submitted). arXiv:1908.08238

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