A simple strategy to assess the error in the numerical wave number of the finite element solution of the Helmholtz equation

Abstract

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,

Goal-oriented adaptivity using unconventional error representations for the 1D Helmholtz equation

“…The idea behind that the tilde-version of any error is smaller seems to be that the new approach softens (or eliminates) the pollution effect of the Helmholtz equation. The pollution effect of the Helmholtz equation is discussed in [6,43,44] for instance. This observation may explain why we obtain sharper estimates using e h .…”

Goal-oriented h-adaptivity for the Helmholtz equation: error estimates, local indicators and refinement strategies

“…A simple and effective strategy for guiding goal-oriented adaptive procedures has been presented, based on the postprocessing techniques introduced in [24,25]. Two different representations of the error in the quantities of interest have been studied which provide similar results.…”

“…Since the approximations u * and ψ * are constructed using a constrained least-squares technique, the estimates for the error e * and ε * vanish at the nodes of the coarse mesh, yielding crude approximations if the solutions present large dispersion errors. In the examples, the influence of the dispersion error in the estimates for the quantity of interest is analyzed using the estimates for the dispersion error introduced in [24,25]. These estimates are denoted by E e and E ε for the primal and adjoint problems respectively.…”

“…In this work, the strategies presented in [24,25] are used to recover the enhanced solutions u * and ψ * from u H and ψ H respectively. A simple and inexpensive post-processing technique is used to recover the approximations u * and ψ * of u and ψ in a finer reference mesh of associated characteristic mesh size h H .…”

“…and η e exp = R D (e * exp ) denote the estimates of the linear contribution to the error in the quantity of interest η := O (e) obtained by using the post-processing strategy described in [24,25]. The subindices exp and pol indicate the kind of approximation used in the least squares fitting: either polynomial both for the real and imaginary part of the solution or a complex-exponential fitting (polynomial fitting for the logarithm of the modulus and for the angle).…”