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Surface finite element approximation of parabolic SPDEs with Whittle--Matérn noise

Øyvind Stormark Auestad, Geir-Arne Fuglstad, Annika Lang

TL;DR

This work addresses numerical approximation of linear parabolic SPDEs on surfaces with additive Whittle--Matérn noise, by a fully discrete surface FEM that discretizes both the operator and the noise via a sinc quadrature. The main contribution is a rigorous convergence analysis giving strong and pathwise error rates that match flat-domain analogues, uniform in general (non-selfadjoint) elliptic operators and arbitrary smooth surfaces. The approach combines backward Euler time stepping with a quadrature-based fractional operator approximation, enabling efficient sampling of Whittle--Matérn fields on surfaces. Numerical experiments on spheres validate the theoretical rates and demonstrate practical applicability to spatio-temporal stochastic models on curved domains.

Abstract

We propose and analyse a new type of fully discrete surface finite element approximation of a class of linear parabolic stochastic evolution equations with additive noise. Our discretization uses a surface finite element approximation of the noise, and is tailored for equations with noise having covariance operator defined by (negative powers of) elliptic operators, like Whittle--Matérn random fields. We derive strong and pathwise convergence rates of our approximation, and verify these by numerical experiments.

Surface finite element approximation of parabolic SPDEs with Whittle--Matérn noise

TL;DR

This work addresses numerical approximation of linear parabolic SPDEs on surfaces with additive Whittle--Matérn noise, by a fully discrete surface FEM that discretizes both the operator and the noise via a sinc quadrature. The main contribution is a rigorous convergence analysis giving strong and pathwise error rates that match flat-domain analogues, uniform in general (non-selfadjoint) elliptic operators and arbitrary smooth surfaces. The approach combines backward Euler time stepping with a quadrature-based fractional operator approximation, enabling efficient sampling of Whittle--Matérn fields on surfaces. Numerical experiments on spheres validate the theoretical rates and demonstrate practical applicability to spatio-temporal stochastic models on curved domains.

Abstract

We propose and analyse a new type of fully discrete surface finite element approximation of a class of linear parabolic stochastic evolution equations with additive noise. Our discretization uses a surface finite element approximation of the noise, and is tailored for equations with noise having covariance operator defined by (negative powers of) elliptic operators, like Whittle--Matérn random fields. We derive strong and pathwise convergence rates of our approximation, and verify these by numerical experiments.

Paper Structure

This paper contains 6 sections, 13 theorems, 74 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 examples and verification of convergence rate
  6. Derivation of (\ref{['eq:scheme-basis-coefficients']})

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 \ref{['eq:model-in-intro']} for different $\gamma$ and $\Gamma$. The details are given in Example \ref{['ex:motivation']}.
  • 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 rates.

Theorems & Definitions (31)

  • Lemma 2.1
  • proof
  • Remark 2.3
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • Remark 2.7
  • Lemma 2.8
  • proof
  • Theorem 3.1
  • ...and 21 more