arXiv subjects

Øyvind Stormark Auestad

Publications and source records attributed to Øyvind Stormark Auestad.

4 recordsLinked to original sources

arXiv2025

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

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.

math.NA↗

arXiv2025

An $L^0$-approach to stochastic evolution equations

We introduce a framework for studying pathwise time regularity and numerical approximation of $L^0$-valued stochastic evolution equations. At the core of our framework are two Burkholder--Davis--Gundy type inequalities accommodating Itô integrals with respect to only stochastically integrable processes. The first of these inequalities is formulated in suitable metrics which metrize convergence in probability on the space of integrands and integrals. The second is a modified version, tailored for deriving pathwise properties of the integral. By combining it with a refined version of the Kolmogorov continuity test, we obtain a powerful method for deriving Hölder regularity of Itô integrals in their most general form. Moreover, it provides a simple and powerful way of deriving rates of pathwise convergence of numerical approximations of stochastic evolution equations. Both applications are illustrated for a class of linear parabolic stochastic evolution equations with generalized Whittle--Matérn type noise, and our findings are verified by numerical experiments from this setting.

math.PR↗

arXiv2025

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

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 $2$.

math.NA↗

arXiv2024

Numerical approximation of linear parabolic evolution equations revisited

We obtain rates of convergence of numerical approximations of abstract linear parabolic evolution equations in Banach spaces. Our estimates extend known results from the literature of finite element approximations of parabolic equations to more general equations and numerical approximation methods. As an example, we consider parabolic equations on surfaces and surface finite element approximations.

math.NA↗