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

Abstract

We propose a fourth-order compact scheme on structured meshes for the Helmholtz equation given by R(φ) := f (x) + ∆φ + ξ 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 [48] and in 2D making the choice α = 0.5 we recover the generalized fourth-order compact Pad´e approximation [56, 57] (therein using the parameter γ = 2). We follow [10, 15] for the analysis of this scheme and its performance on square meshes is compared with that of the quasi-stabilized FEM [15]. 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)

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