Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations

Abstract

Stationary systems modelled by elliptic partial differential equations—linear as well as nonlinear—with stochastic coefficients (random fields) are considered. The mathematical setting as a variational problem, existence theorems, and possible discretisations—in particular with respect to the stochastic part—are given and investigated with regard to stability. Different and increasingly sophisticated computational approaches involving both Wiener’s polynomial chaos as well as the Karhunen–Loève expansion are addressed in conjunction with stochastic Galerkin procedures, and stability within the Galerkin framework is established.New and effective algorithms to compute the mean and covariance of the solution are proposed. The similarities and differences with better known Monte Carlo methods are exhibited, as well as alternatives to integration in high-dimensional spaces. Hints are given regarding the numerical implementation and parallelisation. Numerical examples serve as illustration.