Strictly equivalent a~posteriori error estimators for quasi-optimal nonconforming methods

TL;DR

This paper introduces strictly equivalent a posteriori error estimators for quasi-optimal nonconforming finite element methods, providing reliable error bounds for symmetric elliptic problems of second and fourth order.

Contribution

It develops new a posteriori error estimators that are strictly equivalent to the error for nonconforming methods, including cases with complex source terms.

Findings

Estimators are equivalent to the actual error in a strict sense.

Data oscillation parts are bounded by the error and classical oscillations.

Practical estimators are demonstrated on the Crouzeix-Raviart method for Poisson problems.

Abstract

We devise a posteriori error estimators for quasi-optimal nonconforming finite element methods approximating symmetric elliptic problems of second and fourth order. These estimators are defined for all source terms that are admissible to the underlying weak formulations. More importantly, they are equivalent to the error in a strict sense. In particular, their data oscillation part is bounded by the error and, furthermore, can be designed to be bounded by classical data oscillations. The estimators are computable, except for the data oscillation part. Since even the computation of some bound of the oscillation part is not possible in general, we advocate to handle it on a case-by-case basis. We illustrate the practical use of two estimators obtained for the Crouzeix-Raviart method applied to the Poisson problem with a source term that is not a function and its singular part with respect to the Lebesgues measure is not aligned with the mesh.