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<p>In this work, we consider a fully discrete scheme (CN-FEM) for the second-order wave equation in two and three dimensions. The linear finite element method and the Crank–Nicolson method are employed in space and time, respectively. The accuracy of CN-FEM in the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> norm is proven to be first-order in space and second-order in time. To improve spatial accuracy, we introduce the polynomial preserving recovery (PPR) for the gradient of the numerical solution. On mildly structured triangulations, we prove a superconvergence result of the PPR for the gradient of the CN-FEM solution. The proof relies on a supercloseness estimate between the CN-FEM solution and the elliptic projection of the exact solution in the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(H^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> norm, as well as the PPR superconvergence result for elliptic problems. Two numerical experiments are conducted to verify the theoretical results.</p>

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Polynomial Preserving Recovery for a Crank–Nicolson Finite Element Method of the Second-Order Wave Equation

  • 期刊论文

  • 发表日期：

    2026年3月18日

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  • Jing Wu,
  • Jingrun Chen

摘要

In this work, we consider a fully discrete scheme (CN-FEM) for the second-order wave equation in two and three dimensions. The linear finite element method and the Crank–Nicolson method are employed in space and time, respectively. The accuracy of CN-FEM in the \(H^1\) H 1 norm is proven to be first-order in space and second-order in time. To improve spatial accuracy, we introduce the polynomial preserving recovery (PPR) for the gradient of the numerical solution. On mildly structured triangulations, we prove a superconvergence result of the PPR for the gradient of the CN-FEM solution. The proof relies on a supercloseness estimate between the CN-FEM solution and the elliptic projection of the exact solution in the \(H^1\) H 1 norm, as well as the PPR superconvergence result for elliptic problems. Two numerical experiments are conducted to verify the theoretical results.