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<p>Based on the Wachspress generalized barycentric coordinate, we study the coercivity and error analysis of two types of finite volume element schemes for solving the anisotropic diffusion problems over quadrilateral meshes. The first type is the standard method where the line integral is computed exactly, the second type is obtained by employing the trapezoidal rule to approximate the line integral of the first type. For the second type scheme, we suggest a sufficient condition to guarantee the coercivity result, and this condition covers arbitrary quasi-parallelogram and some trapezoidal meshes with full anisotropic diffusion tensor. Moreover, we get the coercivity result of the first type on quasi-parallelogram mesh. Consequently, the coercivity results of these schemes are enriched and improved, which also contributes to the error estimates. An optimal <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H^1\)</EquationSource> </InlineEquation> error estimate of these schemes is proved on quasi-parallelogram mesh. Finally, the numerical experiments are consistent with the theoretical findings.</p>

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Coercivity results of the polygonal finite volume element method based on Wachspress generalized barycentric coordinate over quadrilateral meshes

  • 期刊论文

  • 发表日期：

    2026年1月31日

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  • Yanhui Zhou

摘要

Based on the Wachspress generalized barycentric coordinate, we study the coercivity and error analysis of two types of finite volume element schemes for solving the anisotropic diffusion problems over quadrilateral meshes. The first type is the standard method where the line integral is computed exactly, the second type is obtained by employing the trapezoidal rule to approximate the line integral of the first type. For the second type scheme, we suggest a sufficient condition to guarantee the coercivity result, and this condition covers arbitrary quasi-parallelogram and some trapezoidal meshes with full anisotropic diffusion tensor. Moreover, we get the coercivity result of the first type on quasi-parallelogram mesh. Consequently, the coercivity results of these schemes are enriched and improved, which also contributes to the error estimates. An optimal \(H^1\) error estimate of these schemes is proved on quasi-parallelogram mesh. Finally, the numerical experiments are consistent with the theoretical findings.