Continuous Interior Penalty Finite Element Methods for the Helmholtz Equation with Large Wave Number

Abstract: This paper develops and analyzes some continuous interior penalty finite element methods (CIP-FEMs) using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in two and three dimensions. The novelty of the proposed methods is to use complex penalty parameters with positive imaginary parts. It is proved that, if the penalty parameter is a pure imaginary number $\i\ga$ with $0<\ga\le C$, then the proposed CIP-FEM is stable (hence well-posed) without any mesh constraint. Moreover the method satisfies the error estimates $C_1kh+C_2k^3h^2$ in the $H^1$-norm when $k^3h^2\le C_0$ and $C_1kh+\frac{C_2}{\ga}$ when $k^3h^2> C_0$ and $kh$ is bounded, where $k$ is the wave number, $h$ is the mesh size, and the $C$'s are positive constants independent of $k$, $h$, and $\ga$. Optimal order $L^2$ error estimates are also derived. The analysis is also applied if the penalty parameter is a complex number with positive imaginary part. By taking $\ga\to 0+$, the above estimates are extended to the linear finite element method under the condition $k^3h^2\le C_0$. Numerical results are provided to verify the theoretical findings. It is shown that the penalty parameters may be tuned to greatly reduce the pollution errors.