@Article{CSIAM-AM-1-530,
author = {Li , YonglinZheng , Weiying and Zhu , Xiaopeng},
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 = {<p style="text-align: justify;">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 $λ$&gt;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.</p>},
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}
}