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Analysis of mixed finite element methods on locally refined grids. (English) Zbl 0772.65071

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 P. A. Raviart and J. M. Thomas [Lect. Notes Math. 606, 292-315 (1977; Zbl 0362.65089)], of F. Brezzi, J. Douglas jun., and L. D. Marini [Numer. Math. 47, 217-235 (1985; Zbl 0599.65072)], of F. Brezzi, J. Douglas jun., R. Durán, and M. Fortin [Numer. Math. 51, 237-250 (1987; Zbl 0631.65107)], and of J. Douglas jun. and J. Wang [Calcolo 26, No. 2-4, 121- 133 (1989; Zbl 0714.65084)].

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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References:

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