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<p>The classical arguments employed when obtaining error estimates of Finite Element (FE) discretisations of elliptic problems lead to more restrictive assumptions on the regularity of the exact solution when applied to non-conforming methods. The so-called minimal regularity estimates available in the literature relax some of these assumptions, but are not truly of <i>minimal regularity</i>, since a&#xa0;data oscillation term appears in the error estimate. Employing an approach based on a&#xa0;smoothing operator, we derive for the first time error estimates for Discontinuous Galerkin (DG) type discretisations of non-linear problems with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((p,\delta )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-structure that only assume the natural <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(W^{1,p}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>-regularity of the exact solution, and which do not contain any oscillation terms.</p>

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Quasi-optimal Discontinuous Galerkin discretisations of the \(p\hspace{0.83328pt}\)-Dirichlet problem

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月17日

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  • Jan Blechta,
  • Pablo Alexei Gazca-Orozco,
  • Alex Kaltenbach,
  • Michael Růžička

摘要

The classical arguments employed when obtaining error estimates of Finite Element (FE) discretisations of elliptic problems lead to more restrictive assumptions on the regularity of the exact solution when applied to non-conforming methods. The so-called minimal regularity estimates available in the literature relax some of these assumptions, but are not truly of minimal regularity, since a data oscillation term appears in the error estimate. Employing an approach based on a smoothing operator, we derive for the first time error estimates for Discontinuous Galerkin (DG) type discretisations of non-linear problems with \((p,\delta )\) ( p , δ ) -structure that only assume the natural \(W^{1,p}\) W 1 , p -regularity of the exact solution, and which do not contain any oscillation terms.