A nonnegativity-preserving finite element method for a class of parabolic SPDEs with multiplicative noise

TL;DR

This paper introduces a finite element method for a class of parabolic SPDEs with multiplicative noise that guarantees nonnegativity of solutions, supported by mathematical analysis and numerical experiments.

Contribution

The paper develops a nonnegativity-preserving finite element discretization for parabolic SPDEs with multiplicative noise, ensuring solution positivity unconditionally.

Findings

The proposed method converges and preserves nonnegativity.

Numerical experiments demonstrate advantages over existing methods.

The method is effective for both nonlinear and linear settings.

Abstract

We consider a prototypical parabolic SPDE with finite-dimensional multiplicative noise, which, subject to a nonnegative initial datum, has a unique nonnegative solution. Inspired by well-established techniques in the deterministic case, we introduce a finite element discretization of this SPDE that is convergent and which, subject to a nonnegative initial datum and unconditionally with respect to the spatial discretization parameter, preserves nonnegativity of the numerical solution throughout the course of evolution. We perform a mathematical analysis of this method. In addition, in the associated linear setting, we develop a fully discrete scheme that also preserves nonnegativity, and we present numerical experiments that illustrate the advantages of the proposed method over alternative finite element and finite difference methods that were previously considered in the literature, which do not necessarily guarantee nonnegativity of the numerical solution.