An efficient multiscale finite-element method for frequency-domain seismic wave propagation

The frequency-domain seismic-wave equation, that is, the Helmholtz equation, has many important applications in seismological studies, yet is very challenging to solve, particularly for large geological models. Iterative solvers, domain decomposition, or parallel strategies can partially alleviate the computational burden, but these approaches may still encounter nontrivial difficulties in complex geological models where a sufficiently fine mesh is required to represent the fine-scale heterogeneities. We develop a novel numerical method to solve the frequency-domain acoustic wave equation on the basis of the multiscale finite-element theory. We discretize a heterogeneous model with a coarse mesh and employ carefully constructed high-order multiscale basis functions to form the basis space for the coarse mesh. Solved from medium- and frequency-dependent local problems, these multiscale basis functions can effectively capture the medium's fine-scale heterogeneity and the source's frequency information, leading to a discrete system matrix with a much smaller dimension compared with those from conventional methods. We then obtain an accurate solution to the acoustic Helmholtz equation by solving only a small linear system instead of a large linear system constructed on the fine mesh in conventional methods. We verify our new method using several models of complicated heterogeneities, and the results show that our new multiscale method can solve the Helmholtz equation in complex models with high accuracy and extremely low computational costs.