Seminar in Numerical Analysis: Sébastien Imperiale (Inria — LMS, Ecole Polytechnique, CNRS — Université Paris-Saclay, MΞDISIM)

Stabilisation — using a multiplier strategy — of spurious high frequencies for high-order discretisations of the wave equation

The objective of this work is to propose and analyze numerical schemes to solve transient wave propagation problems that are exponentially stable (i.e. the solution decays to zero exponentially fast). Applications are in data assimilation strategies or the discretisation of absorbing boundary conditions. More precisely the aim of our work is to propose a discretization process that enables to preserve the exponential stability at the discrete level as well as a high order consistency when using a high-order finite element approximation. The main idea is to add to the wave equation a stabilizing term which damps the high-frequency oscillating components of the solutions such as spurious waves. This term is built from a discrete multiplier analysis that proves the exponential stability of the semi-discrete problem at any order without affecting the order of convergence.

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