A Fully Discrete Nonnegativity-Preserving FEM for a Stochastic Heat Equation
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.
Keywords
Cite
@article{arxiv.2602.16508,
title = {A Fully Discrete Nonnegativity-Preserving FEM for a Stochastic Heat Equation},
author = {Owen Hearder and Claude Le Bris and Ana Djurdjevac},
journal= {arXiv preprint arXiv:2602.16508},
year = {2026}
}
Comments
22 pages, 4 figures, the code for the numerical experiments is available via a URL link in section 6; added clarification on the difference of our results to current literature (beginning of section 5), other minor clarifications added and typos corrected