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Finite element approximation of parabolic SPDEs with Whittle--Matérn noise

Øyvind Stormark Auestad, Geir-Arne Fuglstad, Espen Robstad Jakobsen, Annika Lang

Abstract

We propose and analyse a new type of fully discrete finite element approximation of a class of linear stochastic parabolic evolution equations with additive noise. Our discretization differs from previous ones in that we use a finite element approximation of the noise, as opposed to an $L^2$ projection. This approximation is tailored for equations where the noise has covariance operator defined in terms of (negative powers of) elliptic operators, like Whittle--Matérn random fields. Strong convergence rates up to order $2$ in space and $1$ in time are shown and verified by numerical experiments in dimension $1$ and $2$.

Finite element approximation of parabolic SPDEs with Whittle--Matérn noise

Abstract

We propose and analyse a new type of fully discrete finite element approximation of a class of linear stochastic parabolic evolution equations with additive noise. Our discretization differs from previous ones in that we use a finite element approximation of the noise, as opposed to an projection. This approximation is tailored for equations where the noise has covariance operator defined in terms of (negative powers of) elliptic operators, like Whittle--Matérn random fields. Strong convergence rates up to order in space and in time are shown and verified by numerical experiments in dimension and .
Paper Structure (7 sections, 14 theorems, 87 equations, 2 figures)

This paper contains 7 sections, 14 theorems, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Numerical method and convergence results
  4. Proof of Theorem \ref{['theorem:strong-rate']}
  5. Numerical example and verification of convergence rate
  6. Proofs of mild solution existence, and space and time regularity
  7. Derivation of (3.4) and (3.5)

Key Result

Lemma 2.1

Let $\lambda$, $A$ be as above, and $S(\cdot)$ the analytic semigroup generated by $-A$. Then,

Figures (2)

  • Figure 1: Realizations of the SPDE \ref{['ga:eq:NonStatSPDE']} as described in Example \ref{['ga:example3']}.
  • Figure 2: Relative errors. Row 1 and 2 corresponds to Example 1 and 2, while Column 1 and 2 show rates in space and time, respectively. The dashed lines show corresponding theoretical asymptotic rates.

Theorems & Definitions (33)

  • Lemma 2.1
  • proof
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Remark 2.8
  • Lemma 2.9
  • proof
  • Theorem 3.1
  • ...and 23 more