Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Fully Discrete Nonnegativity-Preserving FEM for a Stochastic Heat Equation

Owen Hearder, Claude Le Bris, Ana Djurdjevac

Abstract

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.

A Fully Discrete Nonnegativity-Preserving FEM for a Stochastic Heat Equation

Abstract

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.
Paper Structure (13 sections, 13 theorems, 103 equations, 2 figures)

This paper contains 13 sections, 13 theorems, 103 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on the Continuous Problem
  3. The Mass-Lumped Discretization
  4. Triangulation and finite element space
  5. Mass and Stiffness Matrices
  6. Discretization
  7. The Splitting Scheme and Nonnegativity
  8. Convergence of the Splitting Scheme
  9. The Auxiliary Process
  10. Convergence
  11. Numerical Experiments
  12. Strong Error
  13. Weak Error

Key Result

Theorem 2.1

Under Assumption assumption_1, equation (main_equation) has a unique solution in the sense of Definition mild_solution. Moreover and

Figures (2)

  • Figure 1: Plots for the strong error, (\ref{['strong_error']}), when the nonlinearity is given by (\ref{['sqrt_approx']}) with $\delta=0.1$. Both axes are scaled logarithmically.
  • Figure 2: Plots for the strong error squared when the nonlinearity is $f(x)=\sqrt{\max\{0,x\}}$. The standard error is represented by the green bars and both axes are scaled logarithmically.

Theorems & Definitions (27)

  • Definition 2.1
  • Theorem 2.1
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 17 more