High-order methods as an alternative to using sparse tensor products for stochastic Galerkin FEM

Solutions of random elliptic boundary value problems admit efficient approximations by polynomials on the parameter domain. Each coefficient in such an expansion is a spatially dependent function, and can be approximated within a hierarchy of finite element spaces. If the finite elements are of sufficiently high order, using just a single spatial mesh is predicted to achieve the same convergence rate with respect to the total number of degrees of freedom as sparse tensor product constructions and other multilevel stochastic Galerkin approximations. Numerical computations for an elliptic two-point boundary value problem confirm this and indicate no loss of accuracy for a single-level method compared to using a sparse tensor product with the same total number of degrees of freedom.