Ph.D. Thesis

The subject of the Ph.D. research concerns the study of  Trefftz-type approximation for solving elasto-acoustic wave propagation system by space-time DG method.
This work supported by the Inria – Total S.A. strategic action DIP (Depth Imaging Partnership) under the joint supervision of Hélène Barucq, Henri Calandra and Julien Diaz.

Key words: Trefftz polynomial space, Trefftz-DG method, elasto-acoustic system, space-time mesh.

Discontinuous Finite Element Methods (DG FEM), are basically well-adapted to specifics of wave propagation problems in complex media. However, the high number of degrees of freedom requested for computation, numerically speaking, makes them twice expensive, compared to the standard methods with continuous approximation.

Among the different possible approaches to solve partial differential equations there exist a distinct family of methods (Trefftz-type methods), based on the use of trial functions in the form of exact solution of the governing equations (but not the boundary conditions).
In the sense of its implementation, the TDG methods reduce numerical cost, since the variational formulation contains the surface integrals only. Thus, it makes it possible the exploration of the meshes of different forms, in order to create more realistic application.

Trefftz methods have been widely used for time-harmonic problems, while their application is still limited in time domain, requiring the space-time meshing.

We have developed a theory for solving the coupled elasto-acoustic wave propagation equation. We have studied the existence and the uniqueness of solution, and the unconditional numerical stability of the scheme, based on the error estimates in mesh-dependent norms. We have considered a space-time polynomial basis for numerical discretization. The obtained numerical results are validated with analytical solutions.

Regarding the advantages of the method, following properties have been proven by the numerical tests: high flexibility in the choice of basis functions, better order of convergence, low dispersion.