A Galerkin approach to the numerical solution of stochastic partial differential equations

Michael A. Eckhoff

Abstract

The purpose of this work is to determine how the standard Galerkin methods (Bubnov, Petrov) might be applied to certain recently studied SPDEs. Three theorems in functional analysis form the core of the Galerkin approach: the Riesz Representation Theorem, Friedrichs' Extension Theorem and the Lax-Milgram Theorem. Each deals, in some way, with duality (e.g., functionals and bilinear forms), which is also important in probability theory. The stochastic Galerkin method developed here refers to so-called stochastic extensions of the theorems just cited. Specifically, the focus is on random linear functionals, particularly those which may be used to describe white noise. White noise has been defined in different ways, and the treatment of space-time white noise is even more diverse. A variety of continuous-parameter white noise definitions is surveyed herein. A taxonomy of random linear functionals is presented in order to formalize the stochastic Galerkin technique. The method is then illustrated with a suite of elliptic and parabolic equations. Galerkin and white-noise algorithms are developed and implemented in C ++ . Making use of Hilbert scales, the notion of a white-noise pair is developed. In this context, the Bubnov and Petrov formulations are treated numerically using the concept of a dual Galerkin scheme. It is shown that some stochastic elliptic equations can be solved numerically with the Bubnov technique, even though the weak equation suggests a Petrov implementation. A mass-lumping scheme is devised for a certain class of evolution equations. Using a specialized Petrov method, the test and trial functions generate, for all appearances, a lattice-type model with increased noise strength. The algorithm is based on dual bases--linear B-splines for the trial space and cubic D-splines for the test space--which handle the spatial domain. The resulting partial discretization replicates centered finite-differencing and reduces the continuous-parameter space-time white noise to a decoupled array of (independent) temporal white noise processes. The Ginzburg-Landau equation is used to illustrate this technique. (Abstract shortened by UMI.)