Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation

When applying numerical methods for the computation of stationary waves from the Helmholtz equation, one obtains ‘numerical waves’ that are dispersive also in non-dispersive media. The numerical wave displays a phase velocity that depends on the parameter k of the Helmholtz equation. In dispersion analysis, the phase difference between the exact and the numerical solutions is investigated. In this paper, the authors' recent result on the phase difference for one-dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic. It is then shown that previous error estimates in H¹-norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only. On the other hand, recently proven error estimates for constant resolution contain a pollution term. With certain assumptions on the exact solution, this term is of the order of the phase difference. Thus a link is established between the results of dispersion analysis and the results of numerical analysis. Throughout the paper, the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems. Special attention is given to the performance of the Galerkin method with a higher order of polynomial approximation p(h-p-version).     

A variational multiscale method with subscales on the element boundaries for the Helmholtz equation

In this paper, we apply the variational multiscale method with subgrid scales on the element boundaries to the problem of solving the Helmholtz equation with low‐order finite elements. The expression for the subscales is obtained by imposing the continuity of fluxes across the interelement boundaries. The stabilization parameter is determined by performing a dispersion analysis, yielding the optimal values for the different discretizations and finite element mesh configurations. The performance of the method is compared with that of the standard Galerkin method and the classical Galerkin least‐squares method with very satisfactory results. Some numerical examples illustrate the behavior of the method. Copyright © 2012 John Wiley & Sons, Ltd.     

Conjugate direction methods for helmholtz problems with complex-valued wavenumbers

The convergence behaviour of conjugate direction methods for Helmholtz problems with complex-valued wavenumbers is studied. The model problem is a Galerkin discretization of the scalar Helmholtz equation on square arrays of 2D and 3D, C° linear elements. A series of controlled experiments is performed which use the dimensionless wavenumber and the algebraic size of the system of equations to completely characterize the iterative performance of the solvers. The effects of algebraic size are examined as functions of both mesh refinement and mesh extension within the limits of present-day workstation computing environments. A comparison is drawn between the conjugate direction methods investigated and the equivalent time-domain solution obtained through explicit time-stepping.     

Three‐dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid‐frequency Helmholtz problems

Recently, a discontinuous Galerkin finite element method with plane wave basis functions and Lagrange multiplier degrees of freedom was proposed for the efficient solution in two dimensions of Helmholtz problems in the mid‐frequency regime. In this paper, this method is extended to three dimensions and several new elements are proposed. Computational results obtained for several wave guide and acoustic scattering model problems demonstrate one to two orders of magnitude solution time improvement over the higher‐order Galerkin method. Copyright © 2005 John Wiley & Sons, Ltd.     

Dispersion and pollution of the FEM solution for the Helmholtz equation in one,two and three dimensions

For high wave numbers, the Helmholtz equation suffers the so‐called ‘pollution effect’. This effect is directly related to the dispersion. A method to measure the dispersion on any numerical method related to the classical Galerkin FEM is presented. This method does not require to compute the numerical solution of the problem and is extremely fast. Numerical results on the classical Galerkin FEM (p‐method) is compared to modified methods presented in the literature. A study of the influence of the topology triangles is also carried out. The efficiency of the different methods is compared. The numerical results in two of the mesh and for square elements show that the high order elements control the dispersion well. The most effective modified method is the QSFEM [1,2] but it is also very complicated in the general setting. The residual‐free bubble [3,4] is effective in one dimension but not in higher dimensions. The least‐square method [1,5] approach lowers the dispersion but relatively little. The results for triangular meshes show that the best topology is the ‘criss‐cross’ pattern. Copyright © 1999 John Wiley & Sons, Ltd.     

Stability and accuracy of finite element methods for flow acoustics. I: general theory and application to one‐dimensional propagation

The dispersion properties of finite element models for aeroacoustic propagation based on the convected scalar Helmholtz equation and on the Galbrun equation are examined. The current study focusses on the effect of the mean flow on the dispersion and amplitude errors present in the discrete numerical solutions. A general two‐dimensional dispersion analysis is presented for the discrete problem on a regular unbounded mesh, and results are presented for the particular case of one‐dimensional acoustic propagation in which the wave direction is aligned with the mean flow. The magnitude and sign of the mean flow is shown to have a significant effect on the accuracy of the numerical schemes. Quadratic Helmholtz elements in particular are shown to be much less effective for downstream—as opposed to upstream—propagation, even when the effect of wave shortening or elongation due to the mean flow is taken into account. These trends are also observed in solutions obtained for simple test problems on finite meshes. A similar analysis of two‐dimensional propagation is presented in an accompanying article. Copyright © 2005 John Wiley & Sons, Ltd.     

Higher‐order extensions of a discontinuous Galerkin method for mid‐frequency Helmholtz problems

Recently, a discontinuous Galerkin method with plane wave basis functions and Lagrange multiplier degrees of freedom was proposed for the efficient solution of Helmholtz problems in the mid‐frequency regime. In this paper, this method is extended to higher‐order elements. Performance results obtained for various two‐dimensional problems highlight the advantages of these elements over classical higher‐order Galerkin elements such as Q₂ and Q₄ for the discretization of interior and exterior Helmholtz problems. Copyright © 2004 John Wiley & Sons, Ltd.     

A residual‐based finite element method for the Helmholtz equation

A new residual‐based finite element method for the scalar Helmholtz equation is developed. This method is obtained from the Galerkin approximation by appending terms that are proportional to residuals on element interiors and inter‐element boundaries. The inclusion of residuals on inter‐element boundaries distinguishes this method from the well‐known Galerkin least‐squares method and is crucial to the resulting accuracy of this method. In two dimensions and for regular bilinear quadrilateral finite elements, it is shown via a dispersion analysis that this method has minimal phase error. Numerical experiments are conducted to verify this claim as well as test and compare the performance of this method on unstructured meshes with other methods. It is found that even for unstructured meshes this method retains a high level of accuracy. Copyright © 2000 John Wiley & Sons, Ltd.     

间断Galerkin方法的最优误差限

本文考虑了数性双曲方程的离散Ｇａｌｅｒｋｉｎ方法。如果精确到光滑的并且同是几乎均匀的，证明了对于分层常数元的最优的和超收敛的误差限。     

Estimation of the dispersion error in the numerical wave number of standard and stabilized finite element approximations of the Helmholtz equation

An estimator for the error in the wave number is presented in the context of finite element approximations of the Helmholtz equation. The proposed estimate is an extension of the ideas introduced in Steffens and D'?ez (Comput. Methods Appl. Mech. Engng 2009; 198 :1389–1400). In the previous work, the error assessment technique was developed for standard Galerkin approximations. Here, the methodology is extended to deal also with stabilized approximations of the Helmholtz equation. Thus, the accuracy of the stabilized solutions is analyzed, including also their sensitivity to the stabilization parameters depending on the mesh topology. The procedure builds up an inexpensive approximation of the exact solution, using post‐processing techniques standard in error estimation analysis, from which the estimate of the error in the wave number is computed using a simple closed expression. The recovery technique used in Steffens and Díez (Comput. Methods Appl. Mech. Engng 2009; 198 :1389–1400) is based in a polynomial least‐squares fitting. Here a new recovery strategy is introduced, using exponential (in a complex setup, trigonometric) local approximations respecting the nature of the solution of the wave problem. Copyright © 2011 John Wiley & Sons, Ltd.     

Weight‐adjusted discontinuous Galerkin methods: Matrix‐valued weights and elastic wave propagation in heterogeneous media

Weight‐adjusted inner products are easily invertible approximations to weighted L² inner products. These approximations can be paired with a discontinuous Galerkin (DG) discretization to produce a time‐domain method for wave propagation which is low storage, energy stable, and high‐order accurate for arbitrary heterogeneous media and curvilinear meshes. In this work, we extend weight‐adjusted DG methods to the case of matrix‐valued weights, with the linear elastic wave equation as an application. We present a DG formulation of the symmetric form of the linear elastic wave equation, with upwind‐like dissipation incorporated through simple penalty fluxes. A semidiscrete convergence analysis is given, and numerical results confirm the stability and high‐order accuracy of weight‐adjusted DG for several problems in elastic wave propagation.     

A face‐centred finite volume method for second‐order elliptic problems

This work proposes a novel finite volume paradigm, ie, the face‐centred finite volume (FCFV) method. Contrary to the popular vertex and cell‐centred finite volume methods, the novel FCFV defines the solution on the mesh faces (edges in two dimensions) to construct locally conservative numerical schemes. The idea of the FCFV method stems from a hybridisable discontinuous Galerkin formulation with constant degree of approximation, and thus inheriting the convergence properties of the classical hybridisable discontinuous Galerkin. The resulting FCFV features a global problem in terms of a piecewise constant function defined on the faces of the mesh. The solution and its gradient in each element are then recovered by solving a set of independent element‐by‐element problems. The mathematical formulation of FCFV for Poisson and Stokes equation is derived, and numerical evidence of optimal convergence in two dimensions and three dimensions is provided. Numerical examples are presented to illustrate the accuracy, efficiency, and robustness of the proposed methodology. The results show that, contrary to other finite volume methods, the accuracy of the FCFV method is not sensitive to mesh distortion and stretching. In addition, the FCFV method shows its better performance, accuracy, and robustness using simplicial elements, facilitating its application to problems involving complex geometries in three dimensions.     

A high-wavenumber boundary-element method for an acoustic scattering problem

In this paper we show stability and convergence for a novel Galerkin boundary-element-method approach to the impedance boundary-value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data. This problem models, for example, outdoor sound propagation over inhomogeneous flat terrain. To achieve a good approximation with a relatively low number of degrees of freedom we employ a graded mesh with smaller elements adjacent to discontinuities in impedance, and a special set of basis functions for the Galerkin method so that, on each element, the approximation space consists of polynomials (of degree nu) multiplied by traces of plane waves on the boundary. In the case where the impedance is constant outside an interval [a,b], which only requires the discretization of [a,b], we show theoretically and experimentally that the L(2) error in computing the acoustic field on [a,b] is O(log(nu+3/2)|k(b-a)|M(-(nu+1)), where M is the number of degrees of freedom and k is the wavenumber. This indicates that the proposed method is especially commendable for large intervals or a high wavenumber. In a final section we sketch how the same methodology extends to more general scattering problems.     

Stability and accuracy of finite element methods for flow acoustics. II: Two‐dimensional effects

This is the second of two articles that focus on the dispersion properties of finite element models for acoustic propagation on mean flows. We consider finite element methods based on linear potential theory in which the acoustic disturbance is modelled by the convected Helmholtz equation, and also those based on a mixed Galbrun formulation in which acoustic pressure and Lagrangian displacement are used as discrete variables. The current paper focuses on the effects of numerical anisotropy which are associated with the orientation of the propagating wave to the mean flow and to the grid axes. Conditions which produce aliasing error in the Helmholtz formulation are of particular interest. The 9‐noded Lagrangian element is shown to be superior to the more commonly used 8‐noded serendipity element. In the case of the Galbrun elements, the current analysis indicates that isotropic meshes generally reduce numerical error of triangular elements and that higher order mixed quadrilaterals are generally less effective than an equivalent mesh of lower order triangles. Copyright © 2005 John Wiley & Sons, Ltd.     

Triangular finite elements for the generalized Bessel equation of order m

A new family of triangular finite elements is described, useful for solving the axisymmetric vector Helmholtz equation, and a variety of scalar Helmholtz equation problems which lead to generalized Bessel equations of some order m. This family is similar in principle to the scalar axisymmetric Helmholtz elements derived earlier, but requires both reformulation of its describing equations and corresponding new universal element matrices, for successful computational implementation. The necessary formulation is given in this paper. Matrix elements to the sixth-order inclusive have been calculated and extensively tested computationally.     

Diffraction of short waves modelled using new mapped wave envelope finite and infinite elements

We consider a two‐dimensional wave diffraction problem from a closed body such that the complex progressive wave potential satisfies the Sommerfield condition and the Helmholtz equation. We are interested in the case where the wavelength is much smaller than any other length dimensions of the problem. We introduce new mapped wave envelope infinite elements to model the potential in the far field, and test them for some simple Dirichlet boundary condition problems. They are used in conjuction with wave envelope finite elements developed earlier [1] to model the potential in the near field. An iterative procedure is used in which an initial estimate of the phase is iteratively improved. The iteration scheme, by which the wave envelope and phase are recovered, is described in detail. Copyright © 1999 John Wiley & Sons, Ltd.     

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

We propose a fourth‐order compact scheme on structured meshes for the Helmholtz equation given by R(φ):=f( x )+Δφ+ξ²φ=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((ξ?)⁴), where ξ, ? 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 L² norm, the H¹ semi‐norm and the l^∞ Euclidean norm are done and the pollution effect is found to be small. Copyright © 2010 John Wiley & Sons, Ltd.     

Nodally exact Ritz discretizations of 1D diffusion–absorption and Helmholtz equations by variational FIC and modified equation methods

This article presents the first application of the Finite Calculus (FIC) in a Ritz-FEM variational framework. FIC provides a steplength parametrization of mesh dimensions, which is used to modify the shape functions. This approach is applied to the FEM discretization of the steady-state, one-dimensional, diffusion–absorption and Helmholtz equations. Parametrized linear shape functions are directly inserted into a FIC functional. The resulting Ritz-FIC equations are symmetric and carry a element-level free parameter coming from the function modification process. Both constant- and variable-coefficient cases are studied. It is shown that the parameter can be used to produce nodally exact solutions for the constant coefficient case. The optimal value is found by matching the finite-order modified differential equation (FOMoDE) of the Ritz-FIC equations with the original field equation. The inclusion of the Ritz-FIC models in the context of templates is examined. This inclusion shows that there is an infinite number of nodally exact models for the constant coefficient case. The ingredients of these methods (FIC, Ritz, MoDE and templates) can be extended to multiple dimensions     

The Galerkin boundary element method for exterior problems of 2-D Helmholtz equation with arbitrary wavenumber

Among many efforts put into the problems of eigenvalue for the Helmholtz equation with boundary integral equations, Kleinman proposed a scheme using the simultaneous equations of the Helmholtz integral equation with its boundary normal derivative equation. In this paper, the detailed formulation is given following Kleinman’s scheme. In order to solve the integral equation with hypersingularity, a Galerkin boundary element method is proposed and the idea of regularization in the sense of distributions is applied to transform the hypersingular integral to a weak one. At last, a least square method is applied to solve the overdetermined linear equation system. Several numerical examples testified that the scheme presented is practical and effective for the exterior problems of the 2-D Helmholtz equation with arbitrary wavenumber.