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 Li , and Weiying Zheng , 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 - Yonglin Li , AU - Weiying Zheng , AU - Xiaopeng Zhu , 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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