FEM and CIP-FEM for Helmholtz Equation with High Wave Number and PML truncation
For researchers solving high-frequency Helmholtz scattering problems, this work provides improved theoretical error bounds and a practical method to reduce pollution error in finite element discretizations.
The paper proves that the inf-sup constant for the PML-truncated Helmholtz problem is O(k^{-1}), and establishes preasymptotic error estimates for CIP-FEM and FEM, achieving a convergence rate twice that of previous results. Numerical tests show that tuning the penalty parameter in CIP-FEM can significantly reduce pollution error.
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.