The standard approach for goal oriented error estimation and adaptivity uses an error representation via an adjoint problem, based on the linear functional output representing the quantity of interest. For the assessment of the error in the approximation of the wave number for the Helmholtz problem (also referred to as dispersion or pollution error), this strategy cannot be applied. This is because there is no linear extractor producing the wave number from the solution of the acoustic problem. Moreover, in this context, the error assessment paradigm is reverted in the sense that the exact value of the wave number, κ, is known (it is part of the problem data) and the effort produced in the error assessment technique aims at obtaining the numerical wave number, κ H , as a postprocess of the numerical solution, u H . The strategy introduced in this paper is based on the ideas used in the a priori analysis. A modified equation corresponding to a modified wave number κ m is introduced. Then, the value of κ m such that the modified problem better accommodates the numerical solution u H is taken as the estimate of the numerical wave number κ H . Thus, both global and local versions of the error estimator are proposed. The obtained estimates of the dispersion error match the a priori predicted dispersion error and, in academical examples, the actual values of the error in the wave number.Key words: Wave problems; Helmholtz equation; Error estimation of wave number; Dispersion/pollution error; Goal-oriented adaptivity; Global/local estimates Partially supported by Ministerio de Educación y Ciencia, Grants DPI2007-62395 and BIA2007-66965;Programme Alβan, the European Union Programme of High Level Scholarships for Latin America, scholarship n o E06D100641BR * Corresponding author.Email addresses: lindaura.steffens@upc.edu (Lindaura Maria Steffens), pedro.diez@upc.edu (Pedro Díez).URL: http://www-lacan.upc.edu (Pedro Díez). Preprint submitted to Elsevier 5 December 2008Steffens, L.M., Díez, P., A simple strategy to assess the error in the numerical wave number of the Finite Element solution 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 Díez (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.The effect of the pollution or dispersion error has been extensively addressed in the literature and, concordantly, a priori estimates for the dispersion error have been derived [1][2][3][4][5][6][7]. Also, a posteriori error estimates assessing the accuracy of the finite element approximations of the Helmholtz equation both in global norms or in some specific quantities of interest have been proposed [8][9][10][11][12][13][14][15][16]. However, the issue of measuring the dispersion error of the approximations of the Helmholtz equation using a posteriori error estimates was first addressed in [17].The wave number corresponding to the approximate solution is different from the exact one. The corresponding error is directly related to the dispersion error and it is, according to practitioners, a good measure in order to assess the overall quality of the numerical solution. The problem of assessing the error in the wave number is addressed in [17] for standard finite element (Galerkin) approximations. The proposed error estimation strategy is paradoxical in the sense that, in the error to be assessed, the obvious information is the exact value and all the efforts are devoted to compute the value of the wave number corresponding to the approximate solution. Note that in the usual error estimation business the situation is the opposite: the approximate value is available and the exact value has to be estimated.In practice, standard Galerkin methods are not competitive for high wave numbers because controlling the pollution effect requires using extremely fine meshes. Numerous approaches alleviating this deficiency have been proposed based on modifications of the classical Galerkin approximation [4,[18][19][20]. The Galerkin/least-squares method is one of the most popular techniques. It provides a significant reduction in...
This paper introduces a new goal-oriented adaptive technique based on a simple and effective post-process of the finite element approximations. The goal-oriented character of the estimate is achieved by analyzing both the direct problem and an auxiliary problem, denoted as adjoint or dual problem, which is related to the quantity of interest. Thus, the error estimation technique proposed in this paper would fall into the category of recovery-type explicit residual a posteriori error estimates. The procedure is valid for general linear quantities of interest and it is also extended to non-linear ones. The numerical examples demonstrate the efficiency of the proposed approach and discuss: (1) different error representations, (2) assessment of the dispersion error, and (3) different remeshing criteria.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.