The second order perturbation approach for PDEs on random domains

The present article deals with the solution of boundary value problems on random domains. We apply a second order shape Taylor expansion to approximate the solution's dependence on the random perturbation with third order accuracy in the size of the perturbation’s amplitude. The major advantage of this approach is that we end up with deterministic equations for the solution’s moments. In particular, representations for the first four moments, i.e., expectation, variance, skewness and kurtosis, are derived. These moments are efficiently computable by means of a boundary element method. Numerical results are presented to illustrate the approach.