A Sparse Grid Stochastic Collocation Method for Elliptic Interface Problems with Random Input

In this paper, numerical solutions of elliptic partial differential equations with both random input and interfaces are considered. The random coefficients are piecewise smooth in the physical space and moderately depend on a large number of random variables in the probability space. To relieve the curse of dimensionality, a sparse grid collocation algorithm based on the Smolyak construction is used. The numerical method consists of an immersed finite element discretization in the physical space and a Smolyak construction of the extreme of Chebyshev polynomials in the probability space, which leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. Numerical experiments on two-dimensional domains are also presented. Convergence is verified and compared with the Monte Carlo simulations.