Moment equations for the mixed formulation of the Hodge Laplacian with stochastic loading term

We study the mixed formulation of the stochastic Hodge–Laplace problem defined on an n-dimensional domain D (n≥1), with random forcing term. In particular, we focus on the magnetostatic problem and on the Darcy problem in the three-dimensional case. We derive and analyse the moment equations, that is, the deterministic equations solved by the mth moment (m≥1) of the unique stochastic solution of the stochastic problem. We find stable tensor product finite element discretizations, both full and sparse, and provide optimal order-of-convergence estimates. In particular, we prove the inf–sup condition for sparse tensor product finite element spaces.