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

2023/12/05 by Li, Yonglin, Wu, Haijun
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2312.02476

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 C₁(kh)^p + C₂k(kh)^2p +C₃ E^{\rm PML} under the mesh condition that k^2p+1h^2p is sufficiently small, where k is the wave number, h is the mesh size, and E^{\rm 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.

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