On the convergence of the stochastic Galerkin method for random elliptic partial differential equations

Abstract

In this article we consider elliptic partial differential equations with random coefficients and/or random forcing terms. In the current treatment of such problems by stochastic Galerkin methods it is standard to assume that the random diffusion coefficient is bounded by positive deterministic constants or modeled as lognormal random field. In contrast, we make the significantly weaker assumption that the non-negative random coefficients can be bounded strictly away from zero and infinity by random variables only and may have distributions different from a lognormal one. We show that in this case the standard stochastic Galerkin approach does not necessarily produce a sequence of approximate solutions that converges in the natural norm to the exact solution even in the case of a lognormal coefficient. By using weighted test function spaces we develop an alternative stochastic Galerkin approach and prove that the associated sequence of approximate solutions converges to the exact solution in the natural norm. Hereby, ideas for the case of lognormal coefficient fields from earlier work of Galvis, Sarkis and Gittelson are used and generalized to the case of positive random coefficient fields with basically arbitrary distributions.