Higher-order FEM and CIP-FEM for Helmholtz equation with high wave number and perfectly matched layer truncation

Abstract

The high-frequency Helmholtz equation on the entire space is truncated into a bounded domain using the perfectly matched layer (PML) technique and subsequently, discretized by the higher-order finite element method (FEM) and the continuous interior penalty finite element method (CIP-FEM). By formulating an elliptic problem involving a linear combination of a finite number of eigenfunctions related to the PML differential operator, a wave-number-explicit decomposition lemma is proved for the PML problem, which implies that the PML solution can be decomposed into a non-oscillating elliptic part and an oscillating but analytic part. The preasymptotic error estimates in the energy norm for both the $p$-th order CIP-FEM and FEM are proved to be $C_1(kh{)}^p+C_{2}k(kh{)}^{2p}+C_3{E}^{\mathrm{PML}}$ under the mesh condition that $k^{2p+1}{h}^{2p}$ is sufficiently small, where $k$ is the wave number, $h$ is the mesh size, and $E^{\mathrm{PML}}$ is the PML truncation error which is exponentially small. In particular, the dependences of coefficients $C_j(j=1,2)$ on the source $f$ are improved. Numerical experiments are presented to validate the theoretical findings, illustrating that the higher-order CIP-FEM can greatly reduce the pollution errors.