A note on the generalized Babuška–Brezzi theory: Revisiting the proof of the associated Strang error estimates

Abstract

In this note we simplify the derivation of the error estimates for the generalized Babuška–Brezzi theory with Galerkin schemes defined in terms of approximate bilinear forms and functionals. More precisely, we provide a straight proof that makes no use of any translated continuous or discrete kernel nor of the distance between them, but of suitable upper bounds of the distances of each component of the Galerkin solution to any other member of the respective finite element subspace. In this way, the Strang error estimates are obtained simply by applying the aforementioned bounds along with the triangle inequality, so that they become cleaner and with fully explicit constants. The case in which the discrete bilinear forms can be evaluated at the continuous solution is also considered, which yields the consistency terms to appear separately from the distances to the subspaces, thus allowing the former to be handled independently from the latter.