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A fourth‐order compact scheme for the Helmholtz equation: Alpha‐interpolation of FEM and FDM stencils
Author(s) -
Nadukandi Prashanth,
Oñate Eugenio,
Garcia Julio
Publication year - 2010
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.3043
Subject(s) - mathematics , helmholtz equation , finite element method , mathematical analysis , norm (philosophy) , interpolation (computer graphics) , geometry , boundary value problem , physics , motion (physics) , classical mechanics , political science , law , thermodynamics
We propose a fourth‐order compact scheme on structured meshes for the Helmholtz equation given by R (φ):= f ( x )+Δφ+ξ 2 φ=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 α=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 γ=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 ((ξℓ) 4 ), where ξ, 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 ∞ Euclidean norm are done and the pollution effect is found to be small. Copyright © 2010 John Wiley & Sons, Ltd.