# KAUST-IAMCS Workshop on Multiscale Modeling, Advanced Discretization Techniques, and Simulation of Wave Propagation

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- King Abdullah University of Science and Technology (KAUST)
- Thuwal, Saudi Arabia

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- Haijun Wu, Nanjing University (China)
- Continuous Interior Penalty Finite Element Methods for the Helmholtz Equation with Large Wave Number

### Abstract

This paper develops and analyzes some continuous interior penalty finite element methods (CIP-FEMs) using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in the two and three dimensions. The novelty of the proposed methods is to use complex penalty parameters with positive imaginary parts. It is proved that the proposed CIP-FEMs are stable (hence well-posed) without any mesh constraint and satisfy the error estimates \(C_{1}kh+C_{2}\text{min}\left(k^{3}h^{2},1+\frac{1}{\left|\gamma\right|}\right)\) in the broken \(H^{1}\)-norm when \(kh\) is bounded, where \(\gamma\) is the penalty parameter, \(k\) is the wave number, and \(h\) Optimal order \(L^{2}\) error estimates are also derived. By taking \(\gamma\to0\), the above estimates are extended to the linear finite element methods under the condition that \(k^{3}h^{2}\) is small enough. Numerical results are provided to verify the theoretical findings. It is shown that the penalty parameter may be tuned to greatly reduce the pollution errors.