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():= f (x)+D+n 2 = 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 = 0.5 we recover the generalized fourth-order compact Padé approximation (J. 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((n ) 4 ), where n, represent the wavenumber and the mesh size, respectively. An expression for the parameter 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. 19As the current problem admits a variational principle, naturally, discretization methods based on variational formulations viz. the Galerkin and the Trefftz-Galerkin-type methods have been preferred to other methods. The Galerkin-type methods are domain-based wherein the integral statement involves only the weak form of the governing differential equation and the sub-space of test-functions are assumed to satisfy a priori the kinematic compatibility and essential boundary conditions. The Trefftz-Galerkin-type methods are boundary-based and are formulated using the reciprocal principle wherein the integral statement involves only the kinematic compatibility and essential boundary conditions of the problem and the sub-space of test-functions are assumed to satisfy a priori the governing differential equation [1,2].In the context of the Galerkin-type methods, the finite element method (FEM) is a powerful technique to systematically generate subspaces of test-functions (classically piecewise polynomial spaces). Some of the earlier works on the use of FEM for the numerical solution of the Helmholtz equation can be found in [3-10] and the references cited therein. In [5,7] error estimates were given for the asymptotic (n 2 assumed sufficiently small) and pre-asymptotic (n assumed sufficiently small) cases, respectively. It was shown that for the discrete problem the LBB ‡ constant can be expressed as h = min{| h m −n 2 |/ h m } [6]. Thus, for the continuous problem (visualized as → 0) the LBB constant can be expressed as = min{| m −n 2 |/ m }, which in an average sense implies that is inversely proportional to the wavenumber n, i.e. ∝ n −1 [6,7]. Thus for high wavenumbers and for the case of degeneracy (n → h m ), the LBB constant for the discrete problem tends to be small which in turn leads to a loss of stability. The loss of stability...

Revisiting generalized FEM: a Petrov–Galerkin enrichment based FEM interpolation for Helmholtz problem

“…Otherwise, if k 2 approaches the finite-dimensional eigenvalues it is said to enter a zone of degeneracy, see the related investigation in [9,24,27]. In 3d, in Appendix B for a specific choice of α B 1 we will get A 0 (κ 2 ) > 0 for all κ 2 , which is the improvement compared to the 2d case.…”

A Petrov–Galerkin formulation for the alpha interpolation of FEM and FDM stencils: Applications to the Helmholtz equation

“…The nonstandard compact stencil presented in can be written as Equation with the following definition of S : …”

“…Taking α 1 = α 2 = α , we arrive at a stencil that is the α interpolation of the Galerkin FEM and the classical central FDM stencils, that is, S α , α = (1 − α ) S fem + α S fdm . Choosing α 1 = α 2 = 0.5, we obtain a stencil that is the average of the FEM and FDM stencils in 2D, and it can be shown to be equal to the stencil obtained by the generalized fourth‐order compact Padé approximation (therein using the parameter γ = 2). Likewise, taking α 1 = 0 and α 2 = α , we obtain a stencil that results from the Galerkin FEM using an α ‐interpolated mass matrix M α := (1 − α ) M + α M L .…”

“…The parameter α may be expressed as a generic series expansion in terms of ω as follows: where a m and b m are coefficients independent of ω . Following the approach used in , which was originally presented in , the relative phase error () and local truncation error () along any direction β can be written as …”

“…This paper is a continuation of wherein a simple domain‐based higher‐order compact numerical scheme involving two parameters, namely α 1 and α 2 , was presented for the Helmholtz equation. The stencil obtained by choosing the parameters as distinct, that is, α 1 ≠ α 2 was denoted therein as the ‘nonstandard compact stencil’.…”