Some observations on Babuška and Brezzi theories (Q1402156)

When the error of a finite element approximation is estimated, often the factor \(\|I-P_h \|\) enters with \(P_h\) being a non-trivial idempotent mapping. This also holds when saddle point problems are treated in the framework of Brezzi's splitting theorem. For simplicity the trivial bound \(\|I-P_h \|\leq 1+\|P_h \|\) is often found. The authors observe that in these cases the sharper bound \(\|I-P_h \|= \|P_h \|\) can be used. For the proof of this equality first a reduction to the two dimensional case is performed. Reviewer's remark. The proof is based on an equality that follows from \textit{simple manipulations.} Since the reviewer did not see the simple manipulations, he had to verify it by a different proof.