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This paper develops and analyzes some continuous interior penalty finiteelement methods (CIP-FEMs) using piecewise linear polynomials for the Helmholtzequation with the first order absorbing boundary condition in two and threedimensions. The novelty of the proposed methods is to use complex penaltyparameters with positive imaginary parts. It is proved that, if the penaltyparameter is a pure imaginary number $\i\ga$ with $0<\ga\le C$, then theproposed 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, andthe $C$'s are positive constants independent of $k$, $h$, and $\ga$. Optimalorder $L^2$ error estimates are also derived. The analysis is also applied ifthe penalty parameter is a complex number with positive imaginary part. Bytaking $\ga\to 0+$, the above estimates are extended to the linear finiteelement method under the condition $k^3h^2\le C_0$. Numerical results areprovided to verify the theoretical findings. It is shown that the penaltyparameters may be tuned to greatly reduce the pollution errors.

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