Finite element and difference approximation of some linear stochastic partial differential equations

Abstract

Difference and finite element methods are described, analyzed, and tested for numerical solution of linear parabolic and elliptic SPDEs driven by white noise. Weak and integral formulations of the stochastic partial differential equations are approximated, respectively, by finite element and difference methods. The white noise processes are approximated by piecewise constant random processes to facilitate convergence proofs for the finite element method. Error analyses of the two numerical methods yield estimates of convergence rates. Computational experiments indicate that the two numerical methods have similar accuracy but the finite element method is computationally more efficient than the difference method