A CIP-FEM for High-Frequency Scattering Problem with the Truncated DtN Boundary Condition (CSIAM Transactions on Applied Mathematics, 2020)
A continuous interior penalty finite element method (CIP-FEM) is proposedto solve high-frequency Helmholtz scattering problem by an impenetrable obstaclein two dimensions. To formulate the problem on a bounded domain, a Dirichlet-to-Neumann (DtN) boundary condition is proposed on the outer boundary by truncatingthe 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-truncationproblem converges exponentially to the original scattering problem as $N$ increases.Under the condition that $k^3$$h^2$ is sufficiently small, we prove that the pre-asymptoticerror estimates for the linear CIP-FEM as well as the linear FEM are $C_1$$kh$+$C_2$$k^3$$h^2$.Numerical experiments are presented to validate the theoretical results.