Continuous interior penalty nite element methods for Helmholtz equation with high wave number

Abstract

This paper analyzes some continuous interior penalty nite element method (CIP-FEM) using piecewise linear polynomials for the Helmholtz equation with the rst order absorbing boundary condition in two and three dimensions. It is proved that if the penalty parameter is chosen as a complex number with positive imaginary part γ-γ r +iγi, then the CIP-FEM is absolute stable, i.e., well-possed for any k; h;R > 0, where k is the wave number, h is the mesh size, and R is the diameter of the domain. Furthermore, if |γ r |≤γi≤1, then there exist constants C 0, C 1, C 2 independent of k,h,γ,R, such that the H 1 error is bounded by ( C 1 kh + C 2 k 3 h 2 R ) RM ( f, g ) when k 3 h 2 R ≤ C 0, and by ( C 1 kh + C 2/γi) RM ( f, g ) when k 3 h 2 R > C 0 and kh ≤ 1, where M ( f, g ) := (‖ f ‖ L 2(Ω) + R -1/2‖ g ‖ L 2(Γ)) + R-1| g | H 1/2(Γ). Optimal order L 2 error estimates are also derived. By taking γ=iγi and letting γi → 0+ in the CIP-FEM, stability and error estimates are obtained for the standard FEM under the condition k> 3 h 2 R ≤ C 0. The previous work of the author considered only pure imaginary penalty parameters and did not consider the dependence on R .