Conventional finite-difference frequency-domain (FDFD) methods can describe wave attenuation and velocity dispersion more easily than time-domain methods. However, there are significant challenges associated with computational costs for solving the linear system when frequency-domain methods are applied in models with large dimensions or fine-scale property variations. Direct-iterative solvers and parallel strategies attempt a tradeoff between memory and time costs. We follow the general framework of heterogeneous multiscale method and develop a multiscale FDFD approach to solve the Helmholtz equation with lower memory and time costs. To achieve this, the discrete linear system approximating the Helmholtz equation is constructed on a coarse mesh, making its dimension much smaller than that of conventional methods. The coefficient matrix in the linear system of dimension-reduction captures fine-scale heterogeneity in the media by coupling fine- and coarse-scale meshes. Several test models are used to verify the accuracy of our multiscale method and investigate potential sources of error. Numerical results demonstrate that our method accurately approximates the wavefields of fine-scale solutions at low frequencies of the source, and could produce solutions with small errors by reducing the size of the coarse mesh cells at high frequencies as well. Comparisons of computational costs with conventional FDFD methods show that the proposed multiscale method significantly reduces computation time and memory consumption.
The Coarse Mesh Condensation Multiscale Method for parallel computation of heterogeneous linear structures without scale separation
