The Helmholtz equation with high wave number on the entire space is truncated into a bounded domain using the perfectly matched layer (PML) technique and subsequently discretized by the higher-order finite element method (FEM) and the continuous interior penalty finite element method (CIP-FEM). By formulating an elliptic problem involving a linear combination of a finite number of eigenfunctions related to the PML differential operator, a wave-number-explicit decomposition lemma is proved for the PML problem, which implies that the PML solution can be decomposed into a non-oscillating elliptic part and an oscillating but piecewise smooth part. The preasymptotic error estimates in the energy norm for both the p-th order CIP-FEM and FEM are proved to be <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2959_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="224" /> </InlineMediaObject> <EquationSource Format="TEX">$C_1(kh)^p + C_2k(kh)^{2p} +C_3\mathcal {E}^{\textrm{PML}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <mi>p</mi> </msup> <mo>+</mo> <msub> <mi>C</mi> <mn>2</mn> </msub> <mi>k</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>C</mi> <mn>3</mn> </msub> <msup> <mrow> <mi mathvariant="script">E</mi> </mrow> <mtext>PML</mtext> </msup> </mrow> </math></EquationSource> </InlineEquation> under the mesh condition that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2959_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$k^{2p+1}h^{2p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>k</mi> <mrow> <mn>2</mn> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>h</mi> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> is sufficiently small, where k is the wave number, h is the mesh size, and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2959_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {E}^{\textrm{PML}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">E</mi> </mrow> <mtext>PML</mtext> </msup> </math></EquationSource> </InlineEquation> is the PML truncation error which is exponentially small. In particular, the dependences of coefficients <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2959_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="91" /> </InlineMediaObject> <EquationSource Format="TEX">$C_j~(j=1,2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>j</mi> </msub> <mspace width="3.33333pt" /> <mrow> <mo stretchy="false">(</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> on the source term are made explicitly and improved, which indicate that the pollution errors of both FEM and CIP-FEM depend only on the <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2959_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norm of source. The presented estimation framework is generalized to the Helmholtz PML problems with general scatterers. Numerical experiments are presented to validate the theoretical findings, illustrating that the higher-order CIP-FEM can greatly reduce the pollution errors by selecting appropriate penalty parameters.

Higher-order FEM and CIP-FEM for Helmholtz Equation with High Wave Number and Perfectly Matched Layer Truncation

期刊论文
发表日期：

2025年6月21日

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Yonglin Li,
Haijun Wu

摘要

The Helmholtz equation with high wave number on the entire space is truncated into a bounded domain using the perfectly matched layer (PML) technique and subsequently discretized by the higher-order finite element method (FEM) and the continuous interior penalty finite element method (CIP-FEM). By formulating an elliptic problem involving a linear combination of a finite number of eigenfunctions related to the PML differential operator, a wave-number-explicit decomposition lemma is proved for the PML problem, which implies that the PML solution can be decomposed into a non-oscillating elliptic part and an oscillating but piecewise smooth part. The preasymptotic error estimates in the energy norm for both the p-th order CIP-FEM and FEM are proved to be $C_1(kh)^p + C_2k(kh)^{2p} +C_3\mathcal {E}^{\textrm{PML}}$ $C_{1} {(k h)}^{p} + C_{2} k {(k h)}^{2 p} + C_{3} E^{PML}$ under the mesh condition that $k^{2p+1}h^{2p}$ $k^{2 p + 1} h^{2 p}$ is sufficiently small, where k is the wave number, h is the mesh size, and $\mathcal {E}^{\textrm{PML}}$ $E^{PML}$ is the PML truncation error which is exponentially small. In particular, the dependences of coefficients $C_j~(j=1,2)$ $C_{j} (j = 1, 2)$ on the source term are made explicitly and improved, which indicate that the pollution errors of both FEM and CIP-FEM depend only on the $L^2$ $L^{2}$ -norm of source. The presented estimation framework is generalized to the Helmholtz PML problems with general scatterers. Numerical experiments are presented to validate the theoretical findings, illustrating that the higher-order CIP-FEM can greatly reduce the pollution errors by selecting appropriate penalty parameters.