Element Method For Helmholtz Equation Research Articles

Linear continuous interior penalty finite element method for Helmholtz equation With High Wave Number: One-Dimensional Analysis

This article addresses the properties of continuous interior penalty (CIP) finite element solutions for the Helmholtz equation. The h-version of the CIP finite element method with piecewise linear approximation is applied to a one-dimensional (1D) model problem. We first show discrete well posedness and convergence results, using the imaginary part of the stabilization operator, for the complex Helmholtz equation. Then we consider a method with real valued penalty parameter and prove an error estimate of the discrete solution in the H1-norm, as the sum of best approximation error plus a pollution term that is the order of the phase difference. It is proved that the pollution effect can be eliminated by selecting the penalty parameter appropriately. As a result of this analysis, thorough and rigorous understanding of the error behavior throughout the range of convergence is gained. Numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior. In particular, we give numerical evidence that the optimal penalty parameter obtained in the 1D case also works very well for the CIP-FEM on two-dimensional Cartesian grids.

A Nitsche-type method for Helmholtz equation with an embedded acoustically permeable interface

We propose a new finite element method for Helmholtz equation in the situation where an acoustically permeable interface is embedded in the computational domain. A variant of Nitsche’s method, different from the standard one, weakly enforces the impedance conditions for transmission through the interface. As opposed to a standard finite-element discretization of the problem, our method seamlessly handles a complex-valued impedance function Z that is allowed to vanish. In the case of a vanishing impedance, the proposed method reduces to the classic Nitsche method to weakly enforce continuity over the interface. We show stability of the method, in terms of a discrete Gårding inequality, for a quite general class of surface impedance functions, provided that possible surface waves are sufficiently resolved by the mesh. Moreover, we prove an a priori error estimate under the assumption that the absolute value of the impedance is bounded away from zero almost everywhere. Numerical experiments illustrate the performance of the method for a number of test cases in 2D and 3D with different interface conditions.

The generalized finite element method for Helmholtz equation. Part II: Effect of choice of handbook functions, error due to absorbing boundary conditions and its assessment

In Part I [T. Strouboulis, I. Babuška, R. Hidajat, The generalized finite element method for Helmholtz equation: theory, computation, and open problems, Comput. Methods Appl. Mech. Engrg. 195 (2006) 4711–4731] we introduced the q-version of the generalized finite element method (GFEM) for the Helmholtz equation and we addressed its: (a) pollution error due to the wave number; (b) exponential q-convergence; (c) robustness to perturbations of the mesh, the roundoff and numerical quadrature errors; and (d) a-posteriori error estimation. Here we continue the development of the GFEM for Helmholtz and we address the effects of: (a) alternative handbook functions and mesh types; (b) the error due to the artificial truncation boundary conditions and its assessment. The conclusions are: (1) the employment of plane-wave, wave-band, and Vekua handbook functions lead to equivalent results; and (2) for high q, the most significant component of error may be the one due to the artificial truncation boundary conditions. A rather straightforward approach for assessing this error is proposed.

A locally discontinuous enriched finite element formulation for acoustics

AbstractIn (Comput. Methods Appl. Mech. Eng. 2006, in press) we introduced a discontinuous Galerkin finite element method for Helmholtz equation in which continuity is relaxed locally in the interior of the element. The shape functions associated with interior nodes of the element are bilinear discontinuous bubbles, and the corresponding degrees of freedom can be eliminated at element level by static condensation yielding a global finite element method with the same connectivity of classical C0 Galerkin finite element approximations. Stability is provided by the discontinuous bubbles with appropriate choice of the stabilization parameters related to the weak enforcement of continuity inside each element. In the present work, departing from the stencil obtained by condensation of the bubble degrees of freedom, we build a new strategy for determining the optimal values of these parameters aiming at matching the exact wave number in two different directions. Stability and accuracy of the proposed formulation are demonstrated in several numerical examples. Copyright © 2006 John Wiley & Sons, Ltd.

The generalized finite element method for Helmholtz equation: Theory, computation, and open problems

In this paper we address the generalized finite element method for the Helmholtz equation. We obtain our method by employing the finite element method on Cartesian meshes, which may overlap the boundaries of the problem domain, and by enriching the approximation by plane waves pasted into the finite element basis at each mesh vertex by the partition of unity method. Here we address the q-convergence of the method, where q is the number of plane waves added at each vertex, for the class of smooth (analytic) solutions for which we get better than exponential convergence for sufficiently small h depending on p. An important observation is that we can monitor the accuracy in any computed solution quantity of interest at negligible cost by using q-extrapolation. Our results assume exact integration of all the employed integrals. Further studies are needed to analyze the effects of the numerical integrations, and also the effect of the roundoff errors.

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.