Year: 2020
Author: Yonglin Li, Weiying Zheng, Xiaopeng Zhu
CSIAM Transactions on Applied Mathematics, Vol. 1 (2020), Iss. 3 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/csiam-am.2020-0025
CSIAM Transactions on Applied Mathematics, Vol. 1 (2020), Iss. 3 : pp. 530–560
Published online: 2020-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 31
Keywords: Helmholtz equation high-frequency DtN operator CIP-FEM wave-number-explicit estimates.