Space–time Trefftz-DG approximation for elasto-acoustics

Abstract

Discontinuous finite element methods have proven their numerical accuracy and flexibility, but they are still criticized for requiring high number of degrees of freedom for computation. There have been some advances with Hybridizable Discontinuous Galerkin (HDG) approximations but essentially in the time-harmonic regime, because the time integration requires implicit schemes. Another possibility to explore seems to be Trefftz methods, which distinguish themselves 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 System (AS) and Elastodynamic System (ES) 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 which 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 (EAS). We confirm well-posedness of the problem, based on the a priory 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.

Explicit Polynomial Trefftz-DG Method for Space-Time Elasto-Acoustics

“…As in standard DG methods, a suitable choice of these penalty parameters allows one to prove stability of the overall method. It is shown in [2,3] that they contribute to the accuracy and convergence of the numerical method. We refer to [2,3] for more details on the definition of the numerical fluxes.…”

“…The analysis of well-posedness of (5) is based on the coercivity and continuity estimates of the bilinear and linear forms in mesh-dependent norms [2,3]. The proof is similar to the one given in [10] where the acoustic wave equation is addressed.…”

“…In this section we provide some important steps of implementation of Trefftz-DG formulation (5) and discuss optimization techniques. The complete algorithm with more numerical details can be found in [2,3].…”

Trefftz-Discontinuous Galerkin Approach for Solving Elastodynamic Problem

“…In this section we show the coercivity and continuity estimates proving wellposedness of the obtained Trefftz-DG method for ES in mesh-dependent norms. We refer to the Appendix B in [21] for more details. The analysis is carried out inside the framework developed in [14] for the time-dependent Maxwell problem.…”

“…Thus, it increases the number of degrees of freedom, and as a result -the global numerical cost of the algorithm. We have computed a space-time polynomial basis, using generating exponential functions -local solutions of the initial systems of equations (see [20,21] for more details). Numerically speaking, this basis generates a lower computational bound than the standard trigonometric ones.…”