Comparison of finite difference and finite element methods for simulating two-dimensional scattering of elastic waves

Two-dimensional scattering of elastic waves in a medium containing a circular heterogeneity is investigated with an analytical solution and numerical wave propagation simulations. Different combinations of finite difference methods (FDM) and finite element methods (FEM) are used to numerically solve the elastodynamic wave equations. Finite difference and finite element techniques are applied to approximate both the time and space derivatives and are combined in various ways to provide different numerical algorithms for modeling elastic wave propagation. The results of the different numerical algorithms are compared for simulations of an incident plane P-wave that is scattered by a mechanically weak circular inclusion whereby the diameter of the inclusion is of the same order than the P-wave's wavelength. For this scattering problem an analytical solution is available and used as the reference solution in the comparison of the different numerical algorithms. Staircase-like spatial discretization of the inclusion's circular shape with the finite difference method using a rectangular grid provides accurate velocity and displacement fields close to the inclusion boundary only for very high spatial resolutions. Implicit time integration based on either finite differences or finite elements does not provide computational advantages compared to explicit schemes. The best numerical algorithm in terms of accuracy and computation time for the investigated scattering problem consists of a finite element method in space using an unstructured mesh combined with an explicit finite difference method in time. The computational advantages and disadvantages of the different numerical algorithms are discussed.