Linear continuous interior penalty finite element method for Helmholtz equation With High Wave Number: One-Dimensional Analysis

Abstract

This paper addresses the properties of Continuous Interior Penalty (CIP) finite element solutions for the Helmholtz equation. The h-version of the CIP finite element method with piecewise linear approximation is applied to a one-dimensional model problem. We first show discrete well posedness and convergence results, using the imaginary part of the stabilization operator, for the complex Helmholtz equation. Then we consider a method with real valued penalty parameter and prove an error estimate of the discrete solution in the H 1 -norm, as the sum of best approximation error plus a pollution term that is the order of the phase difference. It is proved that the pollution effect can be eliminated by selecting the penalty parameter appropriately. As a result of this analysis, thorough and rigorous understanding of the error behavior throughout the range of convergence is gained. Numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior. In particular, we give numerical evidence that the optimal penalty parameter obtained in one-dimensional case also works very well for the CIP-FEM on two-dimensional Cartesian grids.

Unique continuation for the Helmholtz equation using stabilized finite element methods

“…where we used the pseudolocality (11) to absorb the second term on the right-hand side by the left-hand side. We have…”

“…The continuous interior penalty stabilization (CIP) was introduced for the Helmholtz problem in [26], where stability was shown in the kh 1 regime, and was subsequently used to obtain error bounds for standard piecewise affine elements when k 3 h 2 1. It was then shown in [11] that, in the one dimensional case, the CIP stabilization can also be used to eliminate the pollution error, provided the penalty parameter is appropriately chosen. When deriving error estimates for the stabilized FEM that we herein introduce, we shall make use of the mild condition kh 1.…”

Pre-asymptotic error analysis of hp-interior penalty discontinuous Galerkin methods for the Helmholtz equation with large wave number

“…For γ j,e (j ≥ 1), their real parts are obtained by a dispersion analysis of CIP-FEM for one dimensional problem (cf. [34,56]), while the image parts are simply chosen from the set {0.001pi, −500 ≤ p ≤ 500} for γ 1,e and the set {0.0001pi, −500 ≤ p ≤ 500} for γ 2,e to minimize the relative error of the IPDG solution in H 1 -seminorm with γ 0,e = 100 for wave number k = 50 and mesh size h = 1/20.…”

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

“…The CIP-FEM, which was first proposed by Douglas and Dupont [30] for elliptic and parabolic problems in 1970's, uses the same approximation space as the FEM but modifies the bilinear form of the FEM by adding a least squares term penalizing the jump of the normal derivative of the discrete solution at mesh interfaces. Recently the CIP-FEM has shown great potential in solving the Helmholtz problem with large wave number [53,54,32,11,14]. It is absolute stable if the penalty parameters are chosen as complex numbers with negative imaginary parts, it satisfies an error bound no larger than that of the FEM under the same mesh condition, its penalty parameters may be tuned to greatly reduce the pollution error, and so on.…”