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<p>In this paper, we apply the already known hybrid high-order (HHO) method to solve a certain class of strongly monotone nonlinear elliptic problems. We prove the well-posedness of the discrete formulation and obtain the expected rates of convergence in the energy norm. Concerning the convergence in the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> norm, we notice that, at least pre-asymptotically, one could observe an optimal rate of convergence on coarse enough meshes. The strategy relies on a duality argument and a technical result that mimics a mean value type property for vector-valued functions. Up to the author’s knowledge, this kind of result has not been proven before for nonlinear elliptic problems. Several computational experiments give us numerical evidence that our <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norm error estimate could be improved.</p>

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An a priori \(L^{2}\)–norm error estimate of the hybrid high-order method for a class of strongly monotone nonlinear elliptic problems

  • 期刊论文

  • 发表日期：

    2026年7月6日

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  • Rommel Bustinza,
  • Jonathan Munguia-La-Cotera

摘要

In this paper, we apply the already known hybrid high-order (HHO) method to solve a certain class of strongly monotone nonlinear elliptic problems. We prove the well-posedness of the discrete formulation and obtain the expected rates of convergence in the energy norm. Concerning the convergence in the \(L^2\) L 2 norm, we notice that, at least pre-asymptotically, one could observe an optimal rate of convergence on coarse enough meshes. The strategy relies on a duality argument and a technical result that mimics a mean value type property for vector-valued functions. Up to the author’s knowledge, this kind of result has not been proven before for nonlinear elliptic problems. Several computational experiments give us numerical evidence that our \(L^2\) L 2 -norm error estimate could be improved.