Analysis of mixed finite element methods on locally refined grids (Q1195906)

Local refinements of grids for the mixed finite element method are discussed for a simple second order elliptic model problem (an absorptionfree source problem with Neumann type boundary conditions). For local refinements of rectangular or triangular grids corresponding, locally corrected, projections \(\hat\Pi_ h\) of the projection \(\Pi_ h\) onto the finite element space are constructed. Here the projections have to satisfy a commutation relation \(Q_ h\nabla \cdot = \nabla \cdot \Pi_ h\), where \(Q_ h\) is a common \(L^ 2\)-projection operator. Stability is discussed and error estimates are derived. The results are finally applied to refinements in the finite element spaces of \textit{P. A. Raviart} and \textit{J. M. Thomas} [Lect. Notes Math. 606, 292-315 (1977; Zbl 0362.65089)], of \textit{F. Brezzi}, \textit{J. Douglas jun.}, and \textit{L. D. Marini} [Numer. Math. 47, 217-235 (1985; Zbl 0599.65072)], of \textit{F. Brezzi}, \textit{J. Douglas jun.}, \textit{R. Durán}, and \textit{M. Fortin} [Numer. Math. 51, 237-250 (1987; Zbl 0631.65107)], and of \textit{J. Douglas jun.} and \textit{J. Wang} [Calcolo 26, No. 2-4, 121- 133 (1989; Zbl 0714.65084)].