Explicit residual-based a posteriori error estimation for finite element discretizations of the Helmholtz equation: Computation of the constant and new measures of error estimator quality

Abstract

This paper continues the study of an explicit residual-based a posteriori error estimator developed for finite element discretizations of the Helmholtz equation, in the context of time-harmonic exterior acoustics problems. In previous papers [1–3] the error estimator was derived and subsequently applied to h-adaptive computations of model problems in two dimensions. In the present paper, a methodology is established for computing error estimates; these error estimates are then analyzed in detail. The error estimates are made possible by computing the (scaling) constant, which exists in error estimators of this type. In our case, this constant is difficult to obtain. An algorithm for computing the constant is described. (It is noted that the error indicators used for adaptive computations do not require knowledge of this constant.) Several measures are used to analyze the quality of the error estimates, providing a complete description of the error estimator in the context of these problems. We compute global effectivity indices, and utilize the local (element-wise) effectivity index in computing three additional measures. The first of these is the robustness index. The others are the modified robustness index and the distribution index, which are introduced for the first time. It is found that the global effectivity index varies with problem parameters, in particular the wave number. Robustness indices also vary due to the local error definition and wave number. Values of the distribution index are much better, however, indicating the suitability of the error estimator in guiding efficient adaptive mesh refinement. The analysis of the error estimator is carried out on relatively coarse meshes, typical of those used in practical engineering computations.