Published in Int. Journal for Numerical Methods in Engineering Vol. 86 (1), pp. 18-46, 2011
doi: 10.1002/nme.3043

Abstract

We propose a fourth‐order compact scheme on structured meshes for the Helmholtz equation given by $R(\phi ):=f(x)+\Delta \phi +\xi 2\phi =0$. The scheme consists of taking the alpha‐interpolation of the Galerkin finite element method and the classical central finite difference method. In 1D, this scheme is identical to the alpha‐interpolation method (J. Comput. Appl. Math. 1982; 8(1):15–19) and in 2D making the choice $\alpha =0.5$ we recover the generalized fourth‐order compact Padé approximation (J. Comput. Phys. 1995; 119:252–270; Comput. Meth. Appl. Mech. Engrg 1998; 163:343–358) (therein using the parameter $\gamma =2$). We follow (SIAM Rev. 2000; 42(3):451–484; Comput. Meth. Appl. Mech. Engrg 1995; 128:325–359) for the analysis of this scheme and its performance on square meshes is compared with that of the quasi‐stabilized FEM (Comput. Meth. Appl. Mech. Engrg 1995; 128:325–359). In particular, we show that the relative phase error of the numerical solution and the local truncation error of this scheme for plane wave solutions diminish at the rate $O((\xi l)^{4})$, where $\xi ,l$ represent the wavenumber and the mesh size, respectively. An expression for the parameter α is given that minimizes the maximum relative phase error in a sense that will be explained in Section 4.5. Convergence studies of the error in the $L^{2}$ norm, the $H^{1}$ semi‐norm and the $l^{\infty }$ Euclidean norm are done and the pollution effect is found to be small.