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Strong convergence of finite element approximations for a fourth-order stochastic pseudo-parabolic equation with additive noise

Suprio Bhar, Mrinmay Biswas, Mangala Prasad

Abstract

In this article, we analyze semi-discrete finite element approximation and full discretization of a fourth-order stochastic pseudo-parabolic equation in a bounded convex polygonal domain driven by additive Wiener noise. We use the finite element method for spatial discretization and the semi-implicit method for temporal discretization, and obtain strong convergence rates with respect to both the spatial and temporal mesh sizes. Numerical experiments are presented to support the theoretical convergence rates.

Strong convergence of finite element approximations for a fourth-order stochastic pseudo-parabolic equation with additive noise

Abstract

In this article, we analyze semi-discrete finite element approximation and full discretization of a fourth-order stochastic pseudo-parabolic equation in a bounded convex polygonal domain driven by additive Wiener noise. We use the finite element method for spatial discretization and the semi-implicit method for temporal discretization, and obtain strong convergence rates with respect to both the spatial and temporal mesh sizes. Numerical experiments are presented to support the theoretical convergence rates.

Paper Structure

This paper contains 8 sections, 7 theorems, 92 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setup and Main Results
  3. Preliminaries
  4. Abstract framework and regularity
  5. Finite element approximations
  6. Proof of Main Results
  7. Numerical Experiments
  8. Conclusion

Key Result

Proposition 2.1

Let $f$ be a globally Lipschitz function, $Q$ be trace class operator satisfying $\|A^{(\beta-1)/2}Q ^{1/2}\|_{HS}< \infty$ for $\beta \geq 0$, and $u_0$ is an $\mathcal{F}_0$ measurable $H^2(\mathcal{O})$-valued random variable. Then there is a unique mild solution $\{u(t)\}_{t \geq 0}$ to the equa

Figures (2)

  • Figure 1: The rate of strong convergence in $L_2$-norm with respect to space discretized parameter
  • Figure 2: The rate of strong convergence in $L_2$-norm with respect to time discretized parameter

Theorems & Definitions (14)

  • Proposition 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Definition 3.1: rockner
  • proof : Proof of Proposition \ref{['prop1']}
  • proof : Proof of Lemma \ref{['lemma1']}
  • proof : Proof of Theorem \ref{['thm1']}
  • ...and 4 more