A Fully Discrete Nonnegativity-Preserving FEM for a Stochastic Heat Equation
For researchers working on numerical methods for stochastic PDEs, this work provides a convergent nonnegativity-preserving scheme, though it is incremental as it extends existing semi-discrete results to a fully discrete setting.
The paper introduces a fully discrete finite element method for a stochastic heat equation that preserves nonnegativity of solutions, and proves its convergence under suitable regularity conditions.
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.