Multi-level Monte Carlo weak Galerkin method for elliptic equations with stochastic jump coefficients

In this paper, we present a multi-level Monte Carlo weak Galerkin method for solving elliptic equations with stochastic jump coefficients. The multi-level Monte Carlo technique balances the spatial approximation error and the sampling error. The weak Galerkin technique is a stable and high-order accurate method which can easily handle deterministic partial differential equations with complex geometries or jump coefficients given by each sample, and this method is also able to capture highly complex solutions exhibiting discontinuities or oscillations with high resolution. Comparing with the standard Monte Carlo method, by using the multi-level Monte Carlo weak Galerkin method, the computational cost can be sharply reduced to log-linear complexity with respect to the degree of freedom in spatial direction. The numerical experiments verify the efficiency of our algorithms.