一个完全离散的非负保持有限元法用于随机热方程
A Fully Discrete Nonnegativity-Preserving FEM for a Stochastic Heat Equation
AI总结：
本文提出一种完全离散的有限元方法，结合质量集中有限元和李-特罗特尔分裂策略，保证非负性并证明收敛性，通过数值实验验证方法的有效性及适用范围。
摘要：
我们考虑一个具有非线性有限秩空间着色乘法噪声的随机热方程，当给定非负初始数据时，该方程具有唯一非负解。受现有完全离散有限差分方案结果的启发，并基于半离散质量集中的有限元近似收敛性分析，提出一种结合质量集中有限元与李-特罗特尔分裂策略的完全离散数值方法。该离散化方法在离散水平上保持非负性，并在合适正则性条件下被证明是收敛的。提供严谨的收敛性分析，强调质量集中的作用确保非负性，以及操作分裂在解耦确定性和随机动力学中的作用。展示数值实验以确认收敛速率和非负性保持。此外，我们还检查了几个超出已建立理论范围的数值例子，旨在探讨所提方法的应用范围和潜在限制。
英文摘要：
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.
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