Space-time approximation of stochastic p-Laplace type systems

Dominic Breit; Martina Hofmanová; Sebastien Loisel

doi:10.1137/20M1334310

Space-time approximation of stochastic p-Laplace type systems

Dominic Breit, Martina Hofmanová, Sebastien Loisel

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Abstract

We consider systems of stochastic evolutionary equations of p-Laplace type. We establish convergence rates for a finite-element-based space-time approximation, where the error is measured in a suitable quasi-norm. Under natural regularity assumptions on the solution, our main result provides linear convergence in space and convergence of order (almost) 1/2 in time. The key ingredient of our analysis is a random time-grid, which allows us to compensate for the lack of time regularity. Our theoretical results are confirmed by numerical experiments.

Original language	English
Pages (from-to)	2218–2236
Number of pages	19
Journal	SIAM Journal on Numerical Analysis
Volume	59
Issue number	4
DOIs	https://doi.org/10.1137/20M1334310
Publication status	Published - 12 Aug 2021

Keywords

Finite element methods
Nonlinear Laplace-type systems
Parabolic stochastic PDEs
Space-time discretization

ASJC Scopus subject areas

Numerical Analysis
Computational Mathematics
Applied Mathematics

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10.1137/20M1334310

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This is an Accepted Manuscript of an article to be published in SIAM Journal on Numerical Analysis. © 2021, Society for Industrial and Applied Mathematics
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title = "Space-time approximation of stochastic p-Laplace type systems",

abstract = "We consider systems of stochastic evolutionary equations of p-Laplace type. We establish convergence rates for a finite-element-based space-time approximation, where the error is measured in a suitable quasi-norm. Under natural regularity assumptions on the solution, our main result provides linear convergence in space and convergence of order (almost) 1/2 in time. The key ingredient of our analysis is a random time-grid, which allows us to compensate for the lack of time regularity. Our theoretical results are confirmed by numerical experiments.",

keywords = "Finite element methods, Nonlinear Laplace-type systems, Parabolic stochastic PDEs, Space-time discretization",

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note = "Funding Information: \ast Received by the editors April 27, 2020; accepted for publication (in revised form) May 6, 2021; published electronically August 12, 2021. https://doi.org/10.1137/20M1334310 Funding: The work of the second author was supported by the German Science Foundation DFG via the Collaborative Research Center grant SFB1283. \dagger Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS, UK (d.breit@hw. ac.uk, S.Loisel@hw.ac.uk). \ddagger University Bielefeld, D-33501, Bielefeld, Germany (hofmanova@math.uni-bielefeld.de). Publisher Copyright: {\textcopyright} 2021 Society for Industrial and Applied Mathematics.",

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T1 - Space-time approximation of stochastic p-Laplace type systems

AU - Breit, Dominic

AU - Hofmanová, Martina

AU - Loisel, Sebastien

N1 - Funding Information: \ast Received by the editors April 27, 2020; accepted for publication (in revised form) May 6, 2021; published electronically August 12, 2021. https://doi.org/10.1137/20M1334310 Funding: The work of the second author was supported by the German Science Foundation DFG via the Collaborative Research Center grant SFB1283. \dagger Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS, UK (d.breit@hw. ac.uk, S.Loisel@hw.ac.uk). \ddagger University Bielefeld, D-33501, Bielefeld, Germany (hofmanova@math.uni-bielefeld.de). Publisher Copyright: © 2021 Society for Industrial and Applied Mathematics.

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N2 - We consider systems of stochastic evolutionary equations of p-Laplace type. We establish convergence rates for a finite-element-based space-time approximation, where the error is measured in a suitable quasi-norm. Under natural regularity assumptions on the solution, our main result provides linear convergence in space and convergence of order (almost) 1/2 in time. The key ingredient of our analysis is a random time-grid, which allows us to compensate for the lack of time regularity. Our theoretical results are confirmed by numerical experiments.

AB - We consider systems of stochastic evolutionary equations of p-Laplace type. We establish convergence rates for a finite-element-based space-time approximation, where the error is measured in a suitable quasi-norm. Under natural regularity assumptions on the solution, our main result provides linear convergence in space and convergence of order (almost) 1/2 in time. The key ingredient of our analysis is a random time-grid, which allows us to compensate for the lack of time regularity. Our theoretical results are confirmed by numerical experiments.

KW - Finite element methods

KW - Nonlinear Laplace-type systems

KW - Parabolic stochastic PDEs

KW - Space-time discretization

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JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

IS - 4

ER -

Space-time approximation of stochastic p-Laplace type systems

Abstract

Keywords

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