Multilevel methods for mixed finite elements in three dimensions. (English) Zbl 0929.65124

Hiptmair, Ralf; Hoppe, Ronald H. W.

Numer. Math. 82, No. 2, 253-279 (1999).

Authors’ summary: We consider second-order scalar elliptic boundary value problems posed over three-dimensional domains and their discretization by means of mixed Raviart-Thomas finite elements [cf. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods (1991; Zbl 0788.73002)]. This leads to saddle point problems featuring a discrete flux vector field as additional unknown.
Following R. E. Ewing and J. Wang [RAIRO, Modélisation Math. Anal. Numér. 26, No. 6, 739-756 (1992; Zbl 0765.65104)], the proposed solution procedure is based on splitting flux into divergence free components and a remainder. It leads to a variational problem involving solenoidal Raviart-Thomas vector fields.
A fast iterative solution method for this problem is presented. It exploits the representation of divergence free vector fields as curls of the H(curl)-conforming finite element functions introduced by J. Nédélec [Numer. Math. 35, 315-341 (1980; Zbl 0416.65069)]. We show that a nodal multilevel splitting of these finite element spaces gives rise to an optimal preconditioner for the solenoidal variational problem: Duality techniques in quotient spaces and modern algebraic multigrid theory are the main tools for the proof.

Reviewer: S.F.McCormick (Boulder)

MSC:

65N55	Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F10	Iterative numerical methods for linear systems
65F35	Numerical computation of matrix norms, conditioning, scaling
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25	Boundary value problems for second-order elliptic equations

Keywords:

second-order elliptic boundary value problems; Raviart-Thomas finite elements; multilevel splitting; preconditioner; algebraic multigrid

Citations:

Zbl 0436.65087; Zbl 0788.73002; Zbl 0765.65104; Zbl 0416.65069