Fast multipole method as an efficient solver for 2D elastic wave surface integral equations

Abstract

The fast multipole method (FMM) is very efficient in solving integral equations. This paper applies the method to solve large solid-solid boundary integral equations for elastic waves in two dimensions. The scattering problem is first formulated with the boundary element method. FMM is then introduced to expedite the solution process. By using the FMM technique, the number of floating-point operations of the matrix-vector multiplication in a standard conjugate gradient algorithm is reduced from O(N ²) to O(N ^1.5), where N is the number of unknowns. The matrix-filling time and the memory requirement are also of the order N ^1.5. The computational complexity of the algorithm is further reduced to O(N ^4/3) by using a ray propagation technique. Numerical results are given to show the accuracy and efficiency of FMM compared to the boundary element method with dense matrix.