Our talk presents recent activities on the numerical approximation of the solution of diffusion equations with random coefficient, which has become a model problem for numerical methods in recent years but also has many direct applications. We focus on diffusion coefficients which are only realization-wise bounded away from zero and infinity, a case which arises in many practical models. We extend recent results on the convergence of sparse grid collocation methods to this setting in the case where a mixed variational formulation is used for the deterministic problem.
In addition, building on previous results on existence and uniqueness of variational formulations for such diffusion coefficients, we present a convergence result for the stochastic Galerkin method in the primal formulation. We also give an example of the non-convergence of the Galerkin method unless certain weighting functions are introduced into the variational formulation, as was also necessary in the existence theory.