<p>A simple second-order reconstruction is proposed for the weak Galerkin finite element method of lowest order where solutions are approximated using piecewise constants on element interiors and facets. The method is known to possess nice approximation properties, such as satisfying inf-sup condition and achieving optimal-order convergence. But the computed solution converges typically at a first-order rate in both <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3001_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> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3001_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$H^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> norm as the mesh is refined. The proposed reconstruction uses only the values of the weak Galerkin approximation on element facets and ensures continuity at facet barycenters. Its construction is simple, problem-independent, and does not require solving any systems. Its second-order convergence in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3001_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> norm is proved for the weak Galerkin approximation of Stokes, Poisson, and linear elasticity problems. A conforming reconstruction is also discussed. The second-order convergence of both reconstructions is verified by numerical results in two and three dimensions.</p>

A Second-Order Reconstruction for the Weak Galerkin Finite Element Method of Lowest Order

期刊论文
发表日期：

2025年7月28日

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Weizhang Huang,
Zhuoran Wang

摘要

A simple second-order reconstruction is proposed for the weak Galerkin finite element method of lowest order where solutions are approximated using piecewise constants on element interiors and facets. The method is known to possess nice approximation properties, such as satisfying inf-sup condition and achieving optimal-order convergence. But the computed solution converges typically at a first-order rate in both $L^2$ $L^{2}$ and $H^1$ $H^{1}$ norm as the mesh is refined. The proposed reconstruction uses only the values of the weak Galerkin approximation on element facets and ensures continuity at facet barycenters. Its construction is simple, problem-independent, and does not require solving any systems. Its second-order convergence in $L^2$ $L^{2}$ norm is proved for the weak Galerkin approximation of Stokes, Poisson, and linear elasticity problems. A conforming reconstruction is also discussed. The second-order convergence of both reconstructions is verified by numerical results in two and three dimensions.