<p>In this paper, we present an Euler-Galerkin scheme for a class of stochastic nonlocal partial differential equations with delay. First, the stochastic dynamic system is analyzed and the regularity of the solution is proved by energy method. Then, the fully-discrete scheme is proposed to solve the system numerically. The <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3042_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> upper bound of the numerical solution is given. Error estimate is fulfilled and the strong rate of convergence is obtained as <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3042_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="106" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {O}(\tau ^{1/2-\delta }+h^{q})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>τ</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>-</mo> <mi>δ</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>h</mi> <mi>q</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <i>q</i> is the approximation order of spatial projection and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3042_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation> is an arbitrarily small positive number. Finally, a few numerical experiments are presented to show the convergence order and the evolution of the numerical solutions.</p>

Strong rate of convergence for an Euler-Galerkin discretization of the stochastic nonlocal partial differential equations with delay

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发表日期：

2025年9月1日

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Shuxun Shi,
Jiaohui Xu,
Wenbin Chen

摘要

In this paper, we present an Euler-Galerkin scheme for a class of stochastic nonlocal partial differential equations with delay. First, the stochastic dynamic system is analyzed and the regularity of the solution is proved by energy method. Then, the fully-discrete scheme is proposed to solve the system numerically. The $L^{2}$ $L^{2}$ upper bound of the numerical solution is given. Error estimate is fulfilled and the strong rate of convergence is obtained as $\mathcal {O}(\tau ^{1/2-\delta }+h^{q})$ $O (τ^{1 / 2 - δ} + h^{q})$ , where q is the approximation order of spatial projection and $\delta $ $δ$ is an arbitrarily small positive number. Finally, a few numerical experiments are presented to show the convergence order and the evolution of the numerical solutions.