Discontinuous Galerkin method for the Reverse Time Migration

Finite element methods are considered as the best discretization method for solving the wave equa-tion, even if they lead to the solution of huge systems with several millions of degrees of freedom, since they use meshes adapted to the domain topography and the boundary conditions are naturally taken into account in the variational formulation. Among the different finite element families, the spectral element one (SEM) [1, 2, 3] is very interesting because it matches a Gauss Lobatto quadra-ture rule which leads to a diagonal mass matrix without hampering the convergence order of the finite element method. However, SEM uses meshes of the domain made of quadrangles in 2D or hexaedra in 3D which are difficult to compute and not always suitable for complex topographies. Moreover, as far as we know, the convergence order of SEM is not always guaranteed when the velocity field is varying inside each element. Recently, Grote et al. [4] applied the IPDG (Interior Penalty Dis-continuous Galerkin) method to the wave equation. This approach is very interesting since it relies on meshes with triangles in 2D or tetrahedra in 3D, which allows to handle the topography of the domain very accurately. Moreover, the resulting discrete scheme remains quasi-explicit because the mass matrix is block-diagonal, with no need to approximate its coefficient by a quadrature rules. The order of convergence is then guaranteed as for SEM and it seems to be preserved with polynomia