Haijun Wu, Nanjing University (China)

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

May 7-8, 2011

King Abdullah University of Science and Technology (KAUST)

Thuwal, Kingdom of Saudi Arabia

Continuous Interior Penalty Finite Element Methods for the Helmholtz Equation with Large Wave Number

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 C₁kh + C₂min(k³h², 1 + 1/|γ|) in the broken H¹-norm when kh is bounded, where γ is the penalty parameter, k is the wave number, and h is the mesh size. Optimal order L² error estimates are also derived. By taking γ → 0, the above estimates are extended to the linear finite element methods under the condition that k³h² 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.