We consider a stochastic heat equation with nonlinear multiplicative finite-dimensional 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.
The development of nonnegativity-preserving numerical schemes for stochastic partial differential equations (SPDEs) is motivated by the fact that many SPDEs describe the evolution of particle densities or other intrinsically nonnegative physical quantities; see, for example, [4,9,11,19]. Ensuring that a numerical discretization retains this structural property is essential for maintaining physical consistency and for achieving stable long-time simulations.
As a first step toward more complex conservative SPDE models, the focus of this work is a stochastic heat equation with nonlinear multiplicative finite-dimensional noise of the form du(t) = ∆u(t)dt + f (u(t)) K k=1 e k dB k t , for t ∈ (0, T ],
Under suitable regularity assumptions, the nonnegativity of solutions to (1) for nonnegative initial data and nonlinearity satisfying f (0) = 0, has been established in [6]. In settings with weaker regularity, nonnegativity preservation can still be guaranteed by comparison principles, such as those in [12,Theorem 2.1] and [16,Theorem 2.5].
To the best of our knowledge, the analysis of nonnegativity preservation for fully discrete numerical schemes for SPDEs has so far been confined to finite difference-based discretizations. For instance, [22] studies a generalization of (1) and establishes stability properties of a fully discrete finite difference method. In a related direction, [1] introduces a Lie-Trotter splitting scheme that preserves nonnegativity for an SPDE driven by space-time white noise, thereby restricting the analysis to one spatial dimension. Additional references can be found in [1].
The use of finite element methods in this context offers several advantages over finite difference approaches. Finite elements naturally align with variational formulations and bring with them a rich numerical analysis framework. Moreover, they accommodate complex geometries, support adaptive mesh refinement, and can be easily implemented using standard finite element software developed for deterministic parabolic equations. These features make finite element methods an appealing choice for extending nonnegativity-preserving discretizations to more general SPDEs.
Nonnegativity-preserving finite element schemes have been extensively studied in the deterministic parabolic setting. Early results were obtained in [14], where a lumped mass method was analysed and geometric conditions on the underlying mesh were identified to guarantee nonnegatvity. Subsequent refinements of the lumped mass approach appeared in [5] and [21], which further clarified the role of mesh structure and its interaction with the discrete maximum principle. The extension of these ideas to the stochastic setting was developed in [10], where the same equation (1) was considered and nonnegativity preservation, as well as strong convergence estimates in the H 1 -norm, were established for a semi-discrete spatial discretization.
The numerical scheme studied in this work, denoted by (U LT,h m ) M m=0 , combines the mass-lumped finite element discretization of [10] with the Lie-Trotter splitting scheme introduced in [1]. This yields a fully discrete approximation of (1). The key idea behind the splitting approach is to decompose the equation ( 1) into easily solved subsystems, in our case a purely stochastic subsystem and a purely deterministic subsystem, that can be solved explicitly. The scheme is given by
where τ = T /M denotes the time step size, the δB k m are increments of the driving Brownian motions B k , the D e k are diagonal matrices containing nodal values of the e k , D g(u LT ,h m ) is the diagonal matrix of nodal values of g(u LT,h m ) where g(s) = f (s)/s. The matrices M L and S denote, respectively, the lumped mass and stiffness matrices associated with the mass lumped finite element method.
The main contributions of this work are as follows:
• We extend one of the two fully discrete schemes proposed in [10] from the linear case f (u) = λu to general Lipschitz nonlinearities. We prove that the resulting method preserves nonnegativity (Theorem 4.1). In addition, we derive uniform second-moment bounds for the L 2 norm of the discrete solution, providing key stability estimates for the subsequent convergence analysis (Theorem 5.1).
• The strong L 2 -convergence of the fully discrete numerical scheme with temporal rate 1 2 -ε (for arbitrary small ε) and spatial rate 1 (see Theorem 5.2 and Corollary 5.1).
• Numerical experiments, performed on a two-dimensional spatial domain, which illustrate the strong L 2convergence temporal rate and an improved spatial rate. Further numerical experiments are presented to investigate the performance of the scheme in regimes not addressed by our theoretical results, offering insight into its potential applicability beyond the proven setting (see Section 6).
The findings of this work represent an initial stage in a broader research program and are not intended as definitive conclusions. We offer a snap
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