Formulations of Neumann-type boundary conditions for boundary value problems in the nonlocal framework are beset with difficulties, some related to the choice of a proper scaling. Here we identify a space-dependent scaling for a nonlocal Neumann operator, for which we prove linear in δ (δ being the radius for the support for the kernel) convergence of the Neumann operator and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13662_2025_3923_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>δ</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{O}(\delta ^{2})$</EquationSource> </InlineEquation> convergence of solutions to their classical counterparts. The pointwise-like convergence of the nonlocal normal operator is cast as a new type of two-scale operator-point convergence, which we call condensated convergence. The results hold for general integrable kernels, a setting which is favored in numerical simulations. We support this analysis with numerical convergence studies using a piecewise linear discontinuous Galerkin discretization and show an <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13662_2025_3923_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>δ</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{O}(\delta ^{2})$</EquationSource> </InlineEquation> rate of convergence of solutions, also exhibiting an <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13662_2025_3923_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{O}(h^{2})$</EquationSource> </InlineEquation> convergence, where h is the mesh size.

On nonlocal problems with Neumann boundary conditions: scaling and convergence for nonlocal operators and solutions

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2025年3月14日

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Michael L. Parks,
Petronela Radu

摘要

Formulations of Neumann-type boundary conditions for boundary value problems in the nonlocal framework are beset with difficulties, some related to the choice of a proper scaling. Here we identify a space-dependent scaling for a nonlocal Neumann operator, for which we prove linear in δ (δ being the radius for the support for the kernel) convergence of the Neumann operator and $O (δ^{2})$ $\mathcal{O}(\delta ^{2})$ convergence of solutions to their classical counterparts. The pointwise-like convergence of the nonlocal normal operator is cast as a new type of two-scale operator-point convergence, which we call condensated convergence. The results hold for general integrable kernels, a setting which is favored in numerical simulations. We support this analysis with numerical convergence studies using a piecewise linear discontinuous Galerkin discretization and show an $O (δ^{2})$ $\mathcal{O}(\delta ^{2})$ rate of convergence of solutions, also exhibiting an $O (h^{2})$ $\mathcal{O}(h^{2})$ convergence, where h is the mesh size.