A comparison of explicit continuous and discontinuous Galerkin methods and finite differences for wave propagation in 3D heterogeneous media.

Modeling and imaging techniques for geophysics are extremely demanding in terms of computational resources. Seismic data attempt to resolve smaller scales and deeper targets in increasingly more complex geological settings. Finite elements enable accurate simulation of time-dependent wave propagation in heterogeneous media. They are more costly than finite-difference methods, but this is compensated by their superior accuracy if the finite-element mesh follows the sharp impedance contrasts and by their improved efficiency if the element size scales with wavelength, hence with the local wave velocity.

The use of the finite-element Galerkin method for the semi-discretization of the second-order wave equation leads to a system of second-order ordinary differential equations that needs to be solved with a suitable time-discretization scheme. In particular, when an explicit time stepping is used, the mass matrix arising
from the spatial discretization has to be inverted at each time step, which has dramatic impact on the efficiency of the scheme. In this work we consider two different finite-element formulations that can be used to overcome the problem. The first consists in using continuous finite elements with mass lumping, the second is the Internal Penalty Discontinuous Galerkin. We analyze the accuracy and efficiency of the two methods and compare their performances in term of CPU time.

Finally we address some realistic 3 dimensional settings and compare the results with those obtained with a finite difference scheme.