Stochastic Galerkin methods for elliptic interface problems with random input

In this work, we consider random elliptic interface problems, namely, the media in elliptic equations have both randomness and interfaces. A Galerkin method using bi-orthogonal polynomials is used to convert the random problem into an uncoupled system of deterministic interface problems. A principle on how to choose the orders of the approximated polynomial spaces is given based on the sensitivity analysis in random spaces, with which the total degree of freedom can be significantly reduced. Then immersed finite element methods are introduced to solve the resulting system. Convergence results are given both theoretically and numerically.