An efficient Monte Carlo interior penalty discontinuous Galerkin method for elastic wave scattering in random media

This paper develops and analyzes an efficient Monte Carlo interior penalty discontinuous Galerkin (MCIP-DG) method for elastic wave scattering in random media. The method is constructed based on a multi-modes expansion of the solution of the governing random partial differential equations. It is proved that the mode functions satisfy a three-term recurrence system of partial differential equations (PDEs) which are nearly deterministic in the sense that the randomness only appears in the right-hand side source terms, not in the coefficients of the PDEs. Moreover, the same differential operator applies to all mode functions. A proven unconditionally stable and optimally convergent IP-DG method is used to discretize the deterministic PDE operator, an efficient numerical algorithm is proposed based on combining the Monte Carlo method and the IP-DG method with the $LU$ direct linear solver. It is shown that the algorithm converges optimally with respect to both the mesh size $h$ and the sampling number $M$, and practically its total computational complexity is only amount to solving very few deterministic elastic Helmholtz equations using the $LU$ direct linear solver. Numerically experiments are also presented to demonstrate the performance and key features of the proposed MCIP-DG method.