Volume 1, Issue 3
A CIP-FEM for High-Frequency Scattering Problem with the Truncated DtN Boundary Condition

Yonglin Li, Weiying Zheng & Xiaopeng Zhu

CSIAM Trans. Appl. Math., 1 (2020), pp. 530-560.

Published online: 2020-09

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  • Abstract

A continuous interior penalty finite element method (CIP-FEM) is proposed to solve high-frequency Helmholtz scattering problem by an impenetrable obstacle in two dimensions. To formulate the problem on a bounded domain, a Dirichlet-to-Neumann (DtN) boundary condition is proposed on the outer boundary by truncating the Fourier series of the original DtN mapping into finite terms. Assuming the truncation order $N≥kR$, where $k$ is the wave number and $R$ is the radius of the outer boundary, then the $H^j$-stabilities, $j$ = 0,1,2, are established for both original and dual problems, with explicit and sharp estimates of the upper bounds with respect to $k$. Moreover, we prove that, when $N≥λkR$ for some $λ$>1, the solution to the DtN-truncation problem converges exponentially to the original scattering problem as $N$ increases. Under the condition that $k^3$$h^2$ is sufficiently small, we prove that the pre-asymptotic error estimates for the linear CIP-FEM as well as the linear FEM are $C_1$$kh$+$C_2$$k^3$$h^2$. Numerical experiments are presented to validate the theoretical results.

  • Keywords

Helmholtz equation, high-frequency, DtN operator, CIP-FEM, wave-number-explicit estimates.

  • AMS Subject Headings

65N12, 65N30, 78A40, 78A45

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CSIAM-AM-1-530, author = {Yonglin and Li and and 9158 and and Yonglin Li and Weiying and Zheng and and 9159 and and Weiying Zheng and Xiaopeng and Zhu and and 9160 and and Xiaopeng Zhu}, title = {A CIP-FEM for High-Frequency Scattering Problem with the Truncated DtN Boundary Condition}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2020}, volume = {1}, number = {3}, pages = {530--560}, abstract = {

A continuous interior penalty finite element method (CIP-FEM) is proposed to solve high-frequency Helmholtz scattering problem by an impenetrable obstacle in two dimensions. To formulate the problem on a bounded domain, a Dirichlet-to-Neumann (DtN) boundary condition is proposed on the outer boundary by truncating the Fourier series of the original DtN mapping into finite terms. Assuming the truncation order $N≥kR$, where $k$ is the wave number and $R$ is the radius of the outer boundary, then the $H^j$-stabilities, $j$ = 0,1,2, are established for both original and dual problems, with explicit and sharp estimates of the upper bounds with respect to $k$. Moreover, we prove that, when $N≥λkR$ for some $λ$>1, the solution to the DtN-truncation problem converges exponentially to the original scattering problem as $N$ increases. Under the condition that $k^3$$h^2$ is sufficiently small, we prove that the pre-asymptotic error estimates for the linear CIP-FEM as well as the linear FEM are $C_1$$kh$+$C_2$$k^3$$h^2$. Numerical experiments are presented to validate the theoretical results.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.2020-0025}, url = {http://global-sci.org/intro/article_detail/csiam-am/18307.html} }
TY - JOUR T1 - A CIP-FEM for High-Frequency Scattering Problem with the Truncated DtN Boundary Condition AU - Li , Yonglin AU - Zheng , Weiying AU - Zhu , Xiaopeng JO - CSIAM Transactions on Applied Mathematics VL - 3 SP - 530 EP - 560 PY - 2020 DA - 2020/09 SN - 1 DO - http://doi.org/10.4208/csiam-am.2020-0025 UR - https://global-sci.org/intro/article_detail/csiam-am/18307.html KW - Helmholtz equation, high-frequency, DtN operator, CIP-FEM, wave-number-explicit estimates. AB -

A continuous interior penalty finite element method (CIP-FEM) is proposed to solve high-frequency Helmholtz scattering problem by an impenetrable obstacle in two dimensions. To formulate the problem on a bounded domain, a Dirichlet-to-Neumann (DtN) boundary condition is proposed on the outer boundary by truncating the Fourier series of the original DtN mapping into finite terms. Assuming the truncation order $N≥kR$, where $k$ is the wave number and $R$ is the radius of the outer boundary, then the $H^j$-stabilities, $j$ = 0,1,2, are established for both original and dual problems, with explicit and sharp estimates of the upper bounds with respect to $k$. Moreover, we prove that, when $N≥λkR$ for some $λ$>1, the solution to the DtN-truncation problem converges exponentially to the original scattering problem as $N$ increases. Under the condition that $k^3$$h^2$ is sufficiently small, we prove that the pre-asymptotic error estimates for the linear CIP-FEM as well as the linear FEM are $C_1$$kh$+$C_2$$k^3$$h^2$. Numerical experiments are presented to validate the theoretical results.

Yonglin Li, Weiying Zheng & Xiaopeng Zhu. (2020). A CIP-FEM for High-Frequency Scattering Problem with the Truncated DtN Boundary Condition. CSIAM Transactions on Applied Mathematics. 1 (3). 530-560. doi:10.4208/csiam-am.2020-0025
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