Strictly equivalent a posteriori estimators for quasi-optimal nonconforming methods: abstract theory and first applications

We devise guidelines for deriving a posteriori error estimators for quasi-optimal nonconforming finite element methods and apply them in the case of symmetric elliptic problems of second order. The resulting 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 handling 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 whose singular part with respect to the Lebesgue measure is not aligned with the mesh.