Space-Time Trefftz - Discontinuous Galerkin Approximation for Elasto-Acoustics

Wave reflection imaging for complex media can be effectively done by using advanced numerical methods (see [19]). In the context of the collaborative research program Depth Imaging Partnership (DIP) between Inria and Total, team-project Magique-3D and Prospective Lab of Houston develop high-order numerical schemes based mostly on discontinuous finite element approximation of wave fields. This technique, known as Discontinuous Galerkin (DG) method, is preferred because it takes into account geometrical and physical features of environment, and it is well-adapted for parallel computation [4, 19]. Recently it has been implemented for coupled elasto-acoustic problems, which led to the development of new propagators in time and frequency domains [6, 34]. However, when comparing to the conventional methods based on continuous approximation, thenumber of degrees of freedom required by DG method to achieve the same accuracy is significantlyhigher. To avoid this difficulty, Hybridizable Discontinuous Galerkin (HDG) methods have been developedand their integration into DIP is under way for both acoustic and elastic domains, with possibilityof numerical coupling (see [22] and references therein). Another idea to explore consists in using Trefftz approximation space, whose elements are themselves discrete local solutions of the Acoustic System (AS) and Elastodynamic System (ES) [18, 32]. By its construction, Trefftz method reduces degrees of freedom, since it requires computing the surface integrals only to build variational formulation. Thus, we may consider the following advantages of Trefftz method compared to the standard ones: better order of convergence, flexibility in the choice of basis functions, low dispersion, incorporation of wave propagation directions in thediscrete space, adaptivity and local space-time mesh refinement [18, 32]. Trefftz type methods have been widely used with time-harmonic formulations by Farhat, Tezaur,Harari, Hetmaniuk (2003 - 2006) (see [15, 31]), Gabard (2007) (see [17]), Badics (2014) (see [3]),Hiptmair, Moiola, Perugia (2011 - 2013) (see [20, 21, 28]) and others, while studies are still limitedfor reproducing temporal phenomena. Only few papers are interested in Maxwell equations in time[14, 23, 24, 30], but they are mostly devoted to a theoretical analysis of the method, showing theconvergence and stability, and numerical tests with plane waves approximation are restricted to1D + time dimensional case.Space-time Trefftz approximation by Lagrange multipliers for the second order formulation of thetransient wave equation was explored in [5, 33]. Trefftz - Discontinuous Galerkin (Trefftz-DG) method for the first-order transient acoustic waveequations in arbitrary space dimensions extending the one-dimensional scheme of Kretzschmar etal. [23] has been introduced in recent paper of Moiola and Perugia (2017) [29]. the authors proposea complete a priori error analysis in both mesh-dependent and mesh-independent norms.In order to move on numerical simulations of geophysical phenomena and to consider more realisticapplication, DIP aims to develop new approximation techniques, which retain DG based methodsbut operates the Trefftz approach.