A comparison of continuous mass‐lumped finite elements with finite differences for 3‐D wave propagation

The finite‐difference method on rectangular meshes is widely used for time‐domain modelling of the wave equation. It is relatively easy to implement high‐order spatial discretization schemes and parallelization. Also, the method is computationally efficient. However, the use of finite elements on tetrahedral unstructured meshes is more accurate in complex geometries near sharp interfaces. We compared the standard eighth‐order finite‐difference method to fourth‐order continuous mass‐lumped finite elements in terms of accuracy and computational cost. The results show that, for simple models like a cube with constant density and velocity, the finite‐difference method outperforms the finite‐element method by at least an order of magnitude. Outside the application area of rectangular meshes, i.e., for a model with interior complexity and topography well described by tetrahedra, however, finite‐element methods are about two orders of magnitude faster than finite‐difference methods, for a given accuracy.