A comparison between higher-order finite elements and finite differences for solving the wave equation

High-order finite elements with mass lumping allow for explicit time stepping when integrating the wave equation. An earlier study suggests that this approach can be used for two-dimensional triangulations, but cannot be extended to tetrahedra. Here, however, a new element for tetrahedra is presented. Finite elements for triangles and tetrahedra are better suited to model irregular surfaces and sharp contrasts in velocity models than standard finite differences on regular cartesian grids. The question is whether or not the superior accuracy of the finite element method allows for a reduction of the number of degrees of freedom that is large enough to balance its higher cost. Here it is shown by a comparison on a simple two-dimensional reflection problem that the higher-order finite-element method is actually more efficient than the standard finite-difference method. In addition, a comparison between finite-element schemes of various order suggests that the higher-order approximations are more efficient than the lower-order ones.