A nonnegativity-preserving finite element method for a class of parabolic SPDEs with multiplicative noise
This work addresses the problem of maintaining physical or mathematical constraints in SPDE simulations for researchers in computational mathematics, though it is incremental as it adapts deterministic techniques to the stochastic case.
The paper tackles the challenge of preserving nonnegativity in numerical solutions for a class of parabolic stochastic partial differential equations (SPDEs) with multiplicative noise, introducing a finite element method that ensures nonnegativity unconditionally and demonstrates advantages over existing methods in numerical experiments.
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.