A Fully Discrete Nonnegativity-Preserving FEM for a Stochastic Heat Equation

TL;DR

This paper introduces a fully discrete finite element method for a stochastic heat equation that preserves nonnegativity and converges under certain conditions, validated by theoretical analysis and numerical experiments.

Contribution

A novel fully discrete FEM combining mass-lumping and operator splitting that guarantees nonnegativity and convergence for stochastic heat equations.

Findings

Method preserves nonnegativity at the discrete level.

Convergence of the method is rigorously established.

Numerical experiments confirm theoretical convergence rates.

Abstract

We consider a stochastic heat equation with nonlinear finite-rank space-coloured multiplicative noise that admits a unique nonnegative solution when given nonnegative initial data. Inspired by existing results for fully discrete finite difference schemes and building on the convergence analysis of semi-discrete mass-lumped finite element approximations, a fully discrete numerical method is introduced that combines mass-lumped finite elements with a Lie-Trotter splitting strategy. This discretization preserves nonnegativity at the discrete level and is shown to be convergent under suitable regularity conditions. A rigorous convergence analysis is provided, highlighting the role of mass lumping in ensuring nonnegativity and of operator splitting in decoupling the deterministic and stochastic dynamics. Numerical experiments are presented to confirm the convergence rates and the preservation of nonnegativity. In addition, we examine several numerical examples outside the scope of the established theory, aiming to explore the range of applicability and potential limitations of the proposed method.