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.