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<p>A new continuous interior penalty finite volume method (CIP-FVM) for the Helmholtz equation with large wave number is proposed. After deriving error estimates of the CIP-FVM for an auxiliary elliptic problem with Robin boundary condition, preasymptotic stability and error estimates of the CIP- FVM in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> norms are proved under the mesh condition that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(k^3h^2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>k</mi> <mn>3</mn> </msup> <msup> <mi>h</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> is sufficiently small, where <i>k</i> is the wave number and <i>h</i> is the mesh size. As a corollary, we also obtain the first preasymptotic error estimates for the finite volume method (FVM). The CIP-FVM is stable for any <i>k</i> and <i>h</i> if the imaginary part of the penalty parameter is positive. Numerical tests are provided to verify the theoretical findings and show that the CIP-FVM can significantly reduce pollution error by selecting appropriate penalty parameters.</p>

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Preasymptotic Error Analysis of FVM and CIP-FVM for the Helmholtz Equation with Large Wave Number

  • 期刊论文

  • 发表日期：

    2026年3月31日

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  • Haohao Lei,
  • Junliang Lv,
  • Haijun Wu

摘要

A new continuous interior penalty finite volume method (CIP-FVM) for the Helmholtz equation with large wave number is proposed. After deriving error estimates of the CIP-FVM for an auxiliary elliptic problem with Robin boundary condition, preasymptotic stability and error estimates of the CIP- FVM in \(H^1\) H 1 and \(L^2\) L 2 norms are proved under the mesh condition that \(k^3h^2\) k 3 h 2 is sufficiently small, where k is the wave number and h is the mesh size. As a corollary, we also obtain the first preasymptotic error estimates for the finite volume method (FVM). The CIP-FVM is stable for any k and h if the imaginary part of the penalty parameter is positive. Numerical tests are provided to verify the theoretical findings and show that the CIP-FVM can significantly reduce pollution error by selecting appropriate penalty parameters.