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<p>In this paper we provide some error estimates for the div least-squares finite element method on elliptic problems. The main contribution is presenting a complete error analysis, which improves the current <i>state-of-the-art</i> results. The error estimates for both the scalar and the flux variables are established by specially designed dual arguments with the help of two projections: elliptic projection and H(div) projection, which are crucial to supercloseness estimates. In most cases, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2882_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\(H^3\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> regularity is omitted to get the optimal convergence rate for vector and scalar unknowns, and most of our results require a lower regularity for the vector variable than the scalar. Moreover, a series of supercloseness results are proved, which are <i>never seen</i> in the previous work of least-squares finite element methods.</p>

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Supercloseness Error Estimates for the Div Least-Squares Finite Element Method on Elliptic Problems

  • 期刊论文

  • 发表日期：

    2025年4月16日

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  • Gang Chen,
  • Fanyi Yang,
  • Zheyuan Zhang

摘要

In this paper we provide some error estimates for the div least-squares finite element method on elliptic problems. The main contribution is presenting a complete error analysis, which improves the current state-of-the-art results. The error estimates for both the scalar and the flux variables are established by specially designed dual arguments with the help of two projections: elliptic projection and H(div) projection, which are crucial to supercloseness estimates. In most cases, \(H^3\) H 3 regularity is omitted to get the optimal convergence rate for vector and scalar unknowns, and most of our results require a lower regularity for the vector variable than the scalar. Moreover, a series of supercloseness results are proved, which are never seen in the previous work of least-squares finite element methods.