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

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.