### A simple strategy to assess the error in the numerical wave number of the Finite Element solution of the Helmholtz equation

**Author (s): ** Steffens, L.M.,Díez, P.

**Journal: ** Computer Methods in Applied Mechanics and Engineering

**Volume:** 198, Issues 15-16

**Pages:** 1389 – 1400

**Date:** 2009

**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*

*The strategy introduced in this paper is based on the ideas*

_{H}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*

*. Thus, both global and local versions of*

_{H }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.

**Bibtex:**

@article{Steffens20091389, author = "Steffens, L.M. and Díez, P.", title = "A simple strategy to assess the error in the numerical wave number of the finite element solution of the Helmholtz equation ", journal = "Computer Methods in Applied Mechanics and Engineering ", volume = "198", number = "15–16", pages = "1389 - 1400", year = "2009", issn = "0045-7825", doi = "http://dx.doi.org/10.1016/j.cma.2008.12.005", url = "http://www.sciencedirect.com/science/article/pii/S004578250800426X", }