Space–time Trefftz-DG approximation for elasto-acoustics

Abstract

ABSTRACTDiscontinuous finite element methods have proven their numerical accuracy and flexibility, but they are criticized for requiring high number of degrees of freedom for computation. There have been some advances with Hybridizable Discontinuous Galerkin approximations, but essentially in the time-harmonic regime, because the time integration requires implicit schemes. Another possibility to explore is Trefftz method, which distinguishes itself by the choice of basis functions: they are local solutions of initial equation. Thus, in case of time-dependent problems, space-time meshes are required. Classical Trefftz-type approximation uses the exact solutions of acoustic and elastodynamic systems taken with different frequencies, in order to obtain better numerical errors. We have computed a polynomial basis using the Taylor expansions of generating exponential functions that are the exact solutions of the initial acoustic and elastodynamic systems. This basis, compared to the classical one based on trigonometric functions, requires less degrees of freedom for the same level of accuracy. In the present work, we develop the theory for coupled elasto-acoustic system. We confirm well-posedness of the problem, based on error estimates in mesh-dependent norms. We consider a space-time polynomial basis to implement numerically the method and to perform some numerical experiments showing promising results.