Discontinuous Galerkin methods for solving Helmholtz elastic wave equations for seismic imaging

International audienceOne of the most used seismic imaging methods is the full wave inversion (FWI) method which is an iterative procedure whose algorithm is the following. The algorithm requires to solve 2N wave equations at each iteration. The number of sources, N, is usually large (about 1000) and the efficiency of the inverse solver is thus directly related to the efficiency of the numerical method used to solve the wave equation. Seismic imaging can be performed in the time domain or in the frequency domain regime. We focus here on the second setting. The drawback of time domain is that it requires to store the solution at each time step of the forward simulation. The difficulties related to frequency domain inversion lie in the solution of huge linear systems, which cannot be achieved today when considering realistic 3D elastic media, even with the progress of high-performance computing. In this context, the goal is to develop new forward solvers that reduce the number of degrees of freedom without hampering the accuracy of the numerical solution.We consider here discontinuous Galerkin (DG) methods which are more convenient than finite difference methods to handle the topography of the subsurface. Moreover, they are more adapted than continuous Galerkin (CG) methods to deal with hp-adaptivity. This last characteristics is crucial to adapt the mesh to the different regions of the subsurface which is generally highly heterogeneous. Nevertheless, the main drawback of classical DG methods is that they are expensive because they require a large number of degrees of freedom as compared to CG methods on a given mesh. In this work we consider a new class of DG method, the hybridizable DG (HDG) method. Instead of solving a linear system involving the degrees of freedom of all volumic cells of the mesh, the principle of HDG consists in introducing a Lagrange multiplier representing the trace of the numerical solution on each face of the mesh. Hence, it reduces the number of unknowns of the global linear systems and the volumic solution is recovered thanks to a local computation on each element.We compare the performances of the HDG method with those of classical nodal DG methods, and we present our first results using a HDG method for the first-order form of the elastic wave propagation equations for 2D realistic test-cases