Finite Element Based Second Moment Analysis for Elliptic Problems in Stochastic Domains

We present a finite element method for the numerical solution of elliptic boundary value problems on stochastic domains. The method computes, to leading order in the amplitude of the stochastic boundary perturbation relative to an unperturbed, nominal domain, the mean and the variance of the random solution. The variance is computed as the trace of the solution’s two-point correlation which satisfies a deterministic boundary value problem on the tensor product of the nominal domain. This problem is discretized in the sparse tensor product space by a multilevel frame generated by standard finite elements. The computational complexity of the resulting approach stays essentially proportional to the number of finite elements required for the discretization of the nominal domain.