<p>The finite element method (FEM) is an important tool for approximating partial differential equations (PDEs). The convection-diffusion-reaction (CDR) equations describe the changes in substances in space and time. In this paper, we propose a <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2556_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\theta\)</EquationSource> </InlineEquation>-FEM framework with Wang-Ball elements for solving the CDR equations. The proposed approach integrates Wang-Ball basis functions for spatial discretization and a <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2556_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\theta\)</EquationSource> </InlineEquation>-difference scheme for temporal discretization. Stability analysis and error estimations in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2556_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^2\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2556_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\(H^1\)</EquationSource> </InlineEquation>-semi, <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2556_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^\infty\)</EquationSource> </InlineEquation> norms are rigorously derived, demonstrating that the method is unconditionally stable for <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2556_Article_IEq7.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">\(\frac{1}{2}\leq\theta\leq 1\)</EquationSource> </InlineEquation>. Numerical experiments validate the theoretical results, showcasing the method’s effectiveness for Wang-Ball elements of degree <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2556_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(q > 3\)</EquationSource> </InlineEquation> in both two- and three-dimensional domains.</p>

A \(\theta\)-difference finite element method for solving convection-diffusion-reaction equations based on the Wang-Ball element

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发表日期：

2025年7月7日

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Lanyin Sun,
Ziwei Dong

摘要

The finite element method (FEM) is an important tool for approximating partial differential equations (PDEs). The convection-diffusion-reaction (CDR) equations describe the changes in substances in space and time. In this paper, we propose a \(\theta\) -FEM framework with Wang-Ball elements for solving the CDR equations. The proposed approach integrates Wang-Ball basis functions for spatial discretization and a \(\theta\) -difference scheme for temporal discretization. Stability analysis and error estimations in \(L^2\) , \(H^1\) -semi, \(L^\infty\) norms are rigorously derived, demonstrating that the method is unconditionally stable for \(\frac{1}{2}\leq\theta\leq 1\) . Numerical experiments validate the theoretical results, showcasing the method’s effectiveness for Wang-Ball elements of degree \(q > 3\) in both two- and three-dimensional domains.