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

TL;DR

This paper introduces an efficient Monte Carlo interior penalty discontinuous Galerkin method for simulating elastic wave scattering in random media, leveraging multi-modes expansion and stable discretization techniques.

Contribution

It develops a novel MCIP-DG approach that reduces computational complexity by solving nearly deterministic PDEs with a stable, convergent discretization, and demonstrates its effectiveness through numerical experiments.

Findings

Method converges optimally with respect to mesh size and sampling number.

Computational complexity is significantly reduced by solving few deterministic equations.

Numerical experiments confirm the method's efficiency and accuracy.

Abstract

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.