Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence rates for a finite volume scheme of the stochastic heat equation

Niklas Sapountzoglou, Aleksandra Zimmermann

Abstract

In this contribution, we provide convergence rates for a finite volume scheme of the stochastic heat equation with multiplicative Lipschitz noise and homogeneous Neumann boundary conditions (SHE). More precisely, we give an error estimate for the $L^2$-norm of the space-time discretization of SHE by a semi-implicit Euler scheme with respect to time and a TPFA scheme with respect to space and the variational solution of SHE. The only regularity assumptions additionally needed is spatial regularity of the initial datum and smoothness of the diffusive term.

Convergence rates for a finite volume scheme of the stochastic heat equation

Abstract

In this contribution, we provide convergence rates for a finite volume scheme of the stochastic heat equation with multiplicative Lipschitz noise and homogeneous Neumann boundary conditions (SHE). More precisely, we give an error estimate for the -norm of the space-time discretization of SHE by a semi-implicit Euler scheme with respect to time and a TPFA scheme with respect to space and the variational solution of SHE. The only regularity assumptions additionally needed is spatial regularity of the initial datum and smoothness of the diffusive term.
Paper Structure (18 sections, 16 theorems, 115 equations, 2 figures, 2 tables)

This paper contains 18 sections, 16 theorems, 115 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. State of the art
  3. Aim of the study
  4. Outline
  5. A Finite Volume scheme for the stochastic heat equation
  6. The stochastic heat equation with multiplicative noise
  7. Admissible meshes and notations
  8. The finite volume scheme
  9. Tools from discrete analysis
  10. Regularity assumptions on the data and main result
  11. Semi-implicit Euler scheme for the stochastic heat equation
  12. Stability estimates for the stochastic heat equation
  13. A regularity result
  14. Stability estimates
  15. Convergence rates
  16. ...and 3 more sections

Key Result

Lemma 2.5

Let $N \in \mathbb{N}$ and $a_n, b_n, \alpha \geq 0$ for all $n \in \{1,...,N\}$. Assume that for every $n \in \{1,...,N\}$ Then, for any $n \in \{1,...,N\}$ we have

Figures (2)

  • Figure 1: First graph: squared $L^2$-error with different time steps. Second graph: squared $L^2$-error with different spatial steps.
  • Figure 2: High number of time steps with $L_{max}=16$ and $L=4$ or $L=8$, respectively.

Theorems & Definitions (40)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3: see EGH00, Definition 9.1
  • Remark 2.4
  • Lemma 2.5: S69, Lemma 1
  • Lemma 2.6: B_CC_HF15, Theorem 3.6, see also Flore21, Lemma 1
  • Remark 2.7: BNSZ23, Remark 2.8
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 30 more