Fast solver for Helmholtz equation using multiscale basis functions
Shubin Fu, Kai Gao · 2015
Summary Conventional finite-element method for solving Helmholtz equation on highly heterogeneous media usually require very finely discretized mesh to accurately represent the medium property variations, and therefore is computationally expensive. We present a continuous Galerkin generalized multiscale finite element method for solving Helmholtz equation in media with variable velocity and mass density on coarse mesh. Instead of using polynomial basis functions, our multiscale method use multiscale functions obtained by multiplying the eigenfunctions from local spectral problems with an appropriately selected multiscale partition of unity to form the approximation space on the coarse mesh. These multiscale basis can efficiently capture the information of fine scale features of the medium without solving the problem on the fine mesh. We show by a numerical example that our multiscale method can greatly reduce the dimension of the coefficient matrix of Helmholtz equation system, as well as the computational time, with very small error.