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<p>Various complex interface shapes are often encountered when solving interface problems numerically, which may causes certain difficulties. This paper is dedicated to the development of a Special Immersed Finite Volume (SIFV) method to deal with elliptic interface problems, even in cases where the interface has a non-trivial geometry. This SIFV method works on unfitted meshes without requiring stabilization terms or penalty parameters. By employing the Aubin-Nitsche technique, an optimal <Emphasis Type="BoldItalic">L</Emphasis><sup><b>2</b></sup> convergence is established for the elliptic interface problem. Numerical tests demonstrate that the SIFV method possesses the following numerical performance: (a) it is capable of achieving second order accuracy in the <Emphasis Type="BoldItalic">L</Emphasis><sup><b>2</b></sup> norm; (b) it is applicable to elliptic interface problems with complex interface geometries; (c) it maintains its convergence order with the mesh refinement.</p>

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\(L^2\) error estimate for solving elliptic interface problems by using a special immersed finite volume method

  • 期刊论文

  • 发表日期：

    2025年9月11日

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  • Kai Liu,
  • Pengfei Zhu

摘要

Various complex interface shapes are often encountered when solving interface problems numerically, which may causes certain difficulties. This paper is dedicated to the development of a Special Immersed Finite Volume (SIFV) method to deal with elliptic interface problems, even in cases where the interface has a non-trivial geometry. This SIFV method works on unfitted meshes without requiring stabilization terms or penalty parameters. By employing the Aubin-Nitsche technique, an optimal L2 convergence is established for the elliptic interface problem. Numerical tests demonstrate that the SIFV method possesses the following numerical performance: (a) it is capable of achieving second order accuracy in the L2 norm; (b) it is applicable to elliptic interface problems with complex interface geometries; (c) it maintains its convergence order with the mesh refinement.