A fourth-order compact scheme for the Helmholtz equation: Alpha-interpolation of FEM and FDM stencils

作者：Nadukandi Prashanth^*; Onate Eugenio; Garcia Julio

来源：International Journal for Numerical Methods in Engineering, 2011, 86(1): 18-46.

DOI：10.1002/nme.3043

• Helmholtz equation
• alpha-interpolation of FEM and FDM
• compact stencils
• dispersion analysis

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 Pade 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 alpha 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(infinity) Euclidean norm are done and the pollution effect is found to be small.

• 出版日期2011-4-8