An eXtended Stochastic Finite Element Method for solving stochastic partial differential equations on random domains

Recently, a new strategy was proposed to solve stochastic partial differential equations on random domains. It is based on the extension to the stochastic framework of the eXtended Finite Element Method (X-FEM). This method leads by a “direct” calculus to an explicit solution in terms of the variables describing the randomness on the geometry. It relies on two major points: the implicit representation of complex geometries using random level-set functions and the use of a Galerkin approximation at both stochastic and deterministic levels. In this article, we detail the basis of this technique, from theoretical and technical points of view. Several numerical examples illustrate the efficiency of this method and compare it to other approaches.