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Local refinement via domain decomposition techniques for mixed finite element methods with rectangular Raviart-Thomas elements. (English) Zbl 0705.65080

Domain decomposition methods for partial differential equations, Proc. 3rd Int. Symp. Houston/TX (USA) 1989, 98-114 (1990).
[For the entire collection see Zbl 0695.00026.]
Mixed finite element methods have been used in a wide variety of applications when high accuracy is desired for both a function and its gradient. The authors study here mixed finite element methods for second- order elliptic equations with Dirichlet boundary conditions and it is observed that Neumann boundary conditions do not change the results in any significant way. The Raviart-Thomas approximating spaces for a rectangular non-uniform partition of the domain induced by local grid refinement is considered. The authors introduce the concept of “slave” nodes in the mixed method, construct the corresponding spaces, and give the finite element approximation to the weak saddle-point formulation of the boundary value problem. It is shown that the constructed finite element spaces on the composite grid satisfy the Babuška-Brezzi condition [I. Babuška: Numer. Math. 16, 322-333 (1971; Zbl 0214.420); F. Brezzi: RAIRO Anal. Numer. 2, 129-151 (1974; Zbl 0338.90047); R. S. Falk and J. E. Osborn: ibid. 14, 249-277 (1980; Zbl 0467.65062)]. An error estimate for the finite element solution is also derived. Computational results for local refinement of rectangular elements are also presented.
Reviewer: H.P.Dikshit

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
35J25 Boundary value problems for second-order elliptic equations