A MULTIGRID METHOD FOR ELASTIC SCATTERING\Lambda

Abstract. A finite element discretization and a multigrid method for an acoustic elastodynamic model are presented. The model problem is a channel with rigid walls and elastic obstacle in the middle, Dirichlet boundary condition for the incoming wave, Neumann boundary conditions at rigid surfaces on the sides, and an absorbing boundary at the outgoing side. The discrete solutions appear to converge as O(h2). The discretization is verified by comparing discrete solutions with known analytic solutions for a modified problem. An efficient V-cycle multigrid algorithm is developed to solve the discrete equations. The method employs smoothing by iterating on the squared system, combined with a block Jacobi step to resolve variables in a neighborhood of the interface. The efficiency of the method is demonstrated on numerical experiments. 1. Introduction. Multigrid methods for the solution of the Helmholtz equation of acoustic in scattering are classical [3, 4]; for more recent development, see [5, 8]. However, multigrid methods for the coupled acoustic-elastic problem do not seem to have been investigated. In this paper, we formulate such method and present preliminary numerical experience from a prototype implementation in MATLAB. 2. Formulation of the problem. Consider situation as in Fig. 2.1. The domain \Omega f is filled with fluid, with elastic obstacle \Omega e, and interface \Gamma i