A posteriori error estimation for finite element solutions of Helmholtz equation. part I: the quality of local indicators and estimators

This paper contains a Þrst systematic analysis of a posteriori estimation for Þnite element solutions of the Helmholtz equation. In this Þrst part, it is shown that the standard a posteriori estimates, based only on local computations, severely underestimate the exact error for the classes of wave numbers and the types of meshes employed in engineering analysis. This underestimation can be explained by observing that the standard error estimators cannot detect one component of the error, the pollution error, which is very signiÞcant at high wave numbers. Here, a rigorous analysis is carried out on a one-dimensional model problem. The analytical results for the residual estimator are illustrated and further investigated by numerical evaluation both for a residual estimator and for the ZZ-estimator based on smoothening. In the second part, reliable a posteriori estimators of the pollution error will be constructed