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

武海军, 南京大学数学系教授; 分别于1992、1995、1999年本、硕博毕业于吉林大学; 后在中科院从事博士后研究工作。曾获新世纪人才资助及2012年江苏省数学杰出成就奖, 在第九届全国计算数学年会上做大会报告, 2015年获国家自然科学基金杰出青年基金。