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<p>We construct and analyze a TraceFEM discretization for the surface biharmonic problem. The method utilizes standard quadratic Lagrange finite element spaces defined on a three-dimensional background mesh and a symmetric <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(C^0\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation> interior penalty formulation posed on a piecewise planar approximation of the surface. Stability is achieved through a combination of surface edge penalties and bulk-facet penalization of gradient and Hessian jumps. We prove optimal first-order convergence in a discrete <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(H^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> norm and quadratic convergence in the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> norm.</p>

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A TraceFEM \(C^0\) Interior Penalty Method for the Surface Biharmonic Equation

  • 期刊论文

  • 发表日期：

    2026年7月9日

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  • Michael Neilan,
  • Hongzhi Wan

摘要

We construct and analyze a TraceFEM discretization for the surface biharmonic problem. The method utilizes standard quadratic Lagrange finite element spaces defined on a three-dimensional background mesh and a symmetric \(C^0\) C 0 interior penalty formulation posed on a piecewise planar approximation of the surface. Stability is achieved through a combination of surface edge penalties and bulk-facet penalization of gradient and Hessian jumps. We prove optimal first-order convergence in a discrete \(H^2\) H 2 norm and quadratic convergence in the \(L^2\) L 2 norm.