A CIP-FEM for High-Frequency Scattering Problem with the Truncated DtN Boundary Condition

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 >= lambda kR for some lambda > 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.