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The Helmholtz scattering problem with high wave number is truncated by theperfectly matched layer (PML) technique and then discretized by the linearcontinuous interior penalty finite element method (CIP-FEM). It is proved thatthe truncated PML problem satisfies the inf--sup condition with inf--supconstant of order $O(k^{-1})$. Stability and convergence of the truncated PMLproblem are discussed. In particular, the convergence rate is twice of theprevious result. The preasymptotic error estimates in the energy norm of thelinear CIP-FEM as well as FEM are proved to be $C_1kh+C_2k^3h^2$ under the meshcondition that $k^3h^2$ is sufficiently small. Numerical tests are provided toillustrate the preasymptotic error estimates and show that the penaltyparameter in the CIP-FEM may be tuned to reduce greatly the pollution error.

FEM and CIP-FEM for Helmholtz Equation with High Wave Number and PML truncation