Numerical schemes for the simulation of seismic wave propagation in frequency domain

National audienceFull Waveform Inversion (FWI) is an imaging technique which is widely used for Seismic Imaging. It is an iterative procedure solving 2N harmonic wave equations at each iteration of the algorithm if N sources are used. The number $N$ is usually large (about 1000) and the efficiency of the whole simulation algorithm is directly related to the efficiency of the numerical method used to solve the wave equations.Seismic imaging can be performed by solving time-dependent wave equations but there is an advantage in considering frequency domain. It is indeed not necessary to store the solution at each time step of the forward simulation. This is interesting because seismic imaging involves very large problems with a lot of data. Memory must then be used with attention. The main drawback lies then in solving large linear systems, which represents a challenging task when considering realistic 3D elastic media, despite the recent advances on high performance numerical linear algebra solvers. In this context, the goal of our study is to develop new solvers based on reduced-size matrices without hampering the accuracy of the numerical solution.We consider Discontinuous Galerkin (DG) methods formulated on fully unstructured meshes, which are more convenient than finite difference methods on cartesian grids to handle the topography of the subsurface. DG methods and classical Finite Element (FE) methods mainly differ from discrete functions which are only piecewise continuous in the case of DG approximation. DG methods are then more suitable than Continuous Galerkin (CG) methods to deal with hp-adaptivity. This is a great advantage to DG method which is thus fully adapted to calculations in highly heterogeneous media.Nevertheless, the main drawback of classical DG methods is that they are more expensive in terms of number of unknowns than classical CG methods, especially when arbitrarily high order interpolation of the field components is used. In this case DG methods lead to larger sparse linear systems with a higher number of globally coupled degrees of freedom as compared to CG methods with a same given mesh. In this work we consider a hybridizable DG (HDG) method. The principle of HDG method consists in introducing a Lagrange multiplier representing the trace of the numerical solution on each face of the mesh cells. This new variable exists only on the faces of the mesh and the unknowns of the problem depend on it. This allows us to reduce the number of unknowns of the global linear system. Now the size of the matrix to be inverted only depends on the number of the faces of the mesh and on the number of the degrees of freedom of each face. It is worth noting that for the classical DG method it depends on the number of the cells of the mesh and on the number of the degrees of freedom of each cell. The solution to the initial problem is then recovered thanks to independent elementwise calculation.We have compared the performance of the HDG method with the one of nodal DG methods for the 2D elastic waves propagation in harmonic domain