FEM and CIP-FEM for Helmholtz Equation with High Wave Number and Perfectly Matched Layer Truncation

Abstract

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 C1kh + C2k 3 h 2 under the mesh condition that k 3 h 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. R 0 J n (kt)f n (t)tdt.(2.24)Note that from [49, 10.21(i)], the zeros of J ν (z) are all real for ν ≥ −1. SinceR is not real and J −n = (−1) n J n for integer n, we have J n (kR) = 0, n ∈ Z.

A CIP-FEM for High-Frequency Scattering Problem with the Truncated DtN Boundary Condition

“…For the Helmholtz equation with exact DtN boundary condition, Melenk and Sauter [26] proved that the linear FE error estimate is u−u FEM h H 1 ≤ C 1 kh+C 2 k 3 h 2 provided that k 2 h is sufficiently small. For impedance boundary condition and PML boundary condition, Wu et al proved the same error estimate by assuming that k 3 h 2 is small enough [11,23,28,30].…”

“…The elliptic projection operator will play an important role in the preasymptotic error analysis [11,23,28,30]. For simplicity, we assume γ e ≡ γ and define…”

“…Recently, the CIP-FEM, which was first proposed by Douglas and Dupont for secondorder elliptic and parabolic equations [10], has shown great advantages in dealing with high-frequency Helmholtz problems [11,23,28,30]. Such as, the CIP-FEM uses the same approximation space as the traditional continuous FEM but modifies the bilinear form by adding a penalty term at mesh interfaces.…”

Perfect absorbing layers for wave scattering problems and related topics

“…However, there exists very limited works on the wavenumber explicit analysis of Helmholtz equations of the PML techniques (see, e.g., [64] for an up-to-date summary of analysis in different aspects of PML). The main purpose of Chapter 3 is to conduct wavenumber explicit analysis for the PAL technique.…”

“…where n is a positive integer. We refer to [19] for the detailed error analysis, and also the very recent work [64] for the insightful wavenumber explicit error estimates.…”