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A multigrid algorithm for the lowest-order Raviart-Thomas mixed triangular finite element method. (English) Zbl 0759.65080
For the discretization of the Dirichlet boundary value problem \(- \text{div}(A\text{ grad}(u))=f\) in \(\Omega\), \(u=0\) on \(\partial\Omega\) the mixed finite element method of P. A. Raviart and J. M. Thomas [Lect. Notes Math. 606, 292-315 (1977; Zbl 0362.65089)] is used. The subject of the paper is the development of a multigrid method for finding an approximate solution of the discretized problem.
The basic idea consists in deriving the equivalence of the Raviart-Thomas method to a certain non-conforming method and constructing a multigrid method for this non-conforming discretization. After the derivation of this equivalence the non-conforming method is analyzed, especially, some discretization error estimates are proved.
A \(W\)-cycle multigrid algorithm for the non-conforming discretization and a full multigrid algorithm are described, and their convergence is proved. Finally, the singular Neumann problem is discussed.
Reviewer: M.Jung (Chemnitz)

MSC:
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
35J25 Boundary value problems for second-order elliptic equations
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