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<p>In this paper, we develop and analyze a mixed finite element method for a nonlinear, higher-order model describing nonlinear wave phenomena and exhibiting important conservation properties. A central goal of our approach is to ensure that these properties are preserved at the discrete level while avoiding the challenges typically encountered when constructing finite element subspaces of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H^2(\Omega )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> as would be required in a standard continuous Galerkin framework. At the continuous level, we establish well-posedness and characterize the solution through energy laws and mass conservation. For the semi-discrete formulation, we derive error estimates in various Bôchner spaces. Furthermore, we establish that the implicit fully discrete scheme is well-posed, converges with optimal order and consistent with both mass conservation and an entropy dissipation law. Finally, we confirm the theoretical findings and conservation properties on a set of benchmark problems.</p>

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A conservative Mixed Finite Element Method for a Regularised Nonlinear Long-Wave Model

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月18日

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  • Ankur Ankur,
  • Andrea Cangiani,
  • Ram Jiwari

摘要

In this paper, we develop and analyze a mixed finite element method for a nonlinear, higher-order model describing nonlinear wave phenomena and exhibiting important conservation properties. A central goal of our approach is to ensure that these properties are preserved at the discrete level while avoiding the challenges typically encountered when constructing finite element subspaces of \(H^2(\Omega )\) H 2 ( Ω ) as would be required in a standard continuous Galerkin framework. At the continuous level, we establish well-posedness and characterize the solution through energy laws and mass conservation. For the semi-discrete formulation, we derive error estimates in various Bôchner spaces. Furthermore, we establish that the implicit fully discrete scheme is well-posed, converges with optimal order and consistent with both mass conservation and an entropy dissipation law. Finally, we confirm the theoretical findings and conservation properties on a set of benchmark problems.