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

TL;DR

This work develops a nonnegativity-preserving finite element method for a prototypical parabolic SPDE with finite-dimensional multiplicative noise. By employing a mass-lumped continuous FE discretization on weakly acute meshes, the authors obtain a semidiscrete scheme that preserves nonnegativity unconditionally in both $h$ and $\Delta t$, and they recast it as an SDE system for the FE coefficients. They prove well-posedness and derive an energy bound for the semi-discrete solution, and establish a nonnegativity principle via It\=o's formula adapted to the mass-lumped setting. For the linear case, they design fully discrete schemes based on exponential-Euler–Milstein splitting, including Strang variants, which preserve nonnegativity unconditionally and exhibit favorable strong convergence properties in numerical experiments, outperforming standard schemes that may produce negatives. The results provide a systematic framework for nonnegativity-preserving numerical approximations of density-type SPDEs with multiplicative noise, with potential impact on models like Dean–Kawasaki and related particle-density equations.

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.

Figures (4)

• Figure 1: Rates of convergence for the error as defined in \ref{['eq:definition-error']} (note that \ref{['eq:definition-error']} is the square of the strong numerical error) of the two-step splitting scheme \ref{['eq:fully-discrete-stepping']}, at fixed mesh size $h=2^{-6}$ when the time step $\Delta t$ varies (left), and at fixed time step $\Delta t=2^{-14}$ when $h$ varies (right). Comparisons are shown with the results obtained using the EMa and EMi schemes. Both graphs use a log-log scale.
• Figure 2: Numerical errors, on a linear scale, for the same schemes and with the same parameters as those in Figure \ref{['fig:loglog_total_error']} above.
• Figure 3: Nonnegativity-preservation: comparison, for two values of the coefficient $\lambda=2$ and $\lambda=4$ for the linear right-hand side $f(u)=\lambda\,u$, of the two-step splitting scheme \ref{['eq:fully-discrete-stepping']} with three other schemes: Euler--Maruyama (EMa), Euler Milstein (EMi) and the stochastic exponential Euler integrator scheme (SEXP) from lord2013stochastic.
• Figure 4: Numerical error of the Strang splitting schemes \ref{['eq:fully-discrete-stepping-2']}, on the left, and \ref{['eq:fully-discrete-stepping-2bis']} on the right, both in Log-Log scale and at fixed $h=2^{-6}$ when the time step $\Delta t$ varies.

Theorems & Definitions (18)

• Definition 2.1
• Theorem 2.1
• Theorem 2.2
• Proposition 3.1
• proof
• Lemma 4.1
• proof
• Theorem 4.1
• proof
• Remark 4.1