AN OPTIMAL PRECONDITIONER FOR
A CLASS OF SADDLE POINT PROBLEMS
WITH A PENALTY TERM
AXEL KLAWONN ?
Abstract. Iterative methods are considered for a class of saddle point problems with a penalty term arising from finite element discretizations of certain elliptic problems. An optimal preconditioner which is independent of the discretization and the penalty parameter is constructed. This approach is then used to design an iterative method with a convergence rate independent of the Lam?e parameters occuring in the equations of linear elasticity.
Key words. mixed finite elements, saddle point problems, penalty term, nearly incompressible materials, elasticity, preconditioned conjugate residual method, domain decomposition, multigrid
AMS(MOS) subject classifications. 65F10, 65N22, 65N30, 65N55, 73V05
1. Introduction. In recent years, modern iterative methods, e.g. domain decomposition and multigrid methods, have been applied to parameter dependent problems arising in solid mechanics; see Braess , Braess and Bl?omer , Jung , and Smith . If the direct approach of (low order) conforming finite elements is used in a pure displacement setting, the phenomenon of locking leads to problems. Locking occurs when a parameter, e.g. the Poisson ratio of a material, approaches a limit. The convergence rate of the iterative method and that of the finite element model deteriorates severely when the limit is approached, e.g. when the Poisson ratio tends to 1/2 in the problem of linear elasticity; see Braess  and Jung ; note that one has to make a distinction between the convergence rate of the finite element model and the convergence rate of the iterative method. This deterioration of the convergence rates can be explained by interpreting locking as a problem of ill conditioning; see Braess , pp. 253-254. For a detailed discussion of the locking phenomenon in the finite element model; see Babu<=ska and Suri .
There are different approaches to overcome the problem of locking in the finite element model; nonconforming finite element methods, reduced/selected integration and a reformulation in terms of a saddle point problem with a penalty term. Most of them can be analyzed as saddle point problems with a penalty term; see Braess , Brenner [11,12], Brenner and Scott , Brezzi and Fortin , and Hughes . For all of these approaches it can be proven, that the finite element solution converges uniformly with regard to the penalty parameter but there is still a difference between these methods as far as the iterative solution of the resulting linear systems is concerned. Thus it was observed in Braess and Bl?omer  that the mixed formulation
? Westf?alische Wilhelms-Universit?at, Institut f?ur Numerische und instrumentelle Mathematik, Einsteinstrasse 62, 48149 M?unster, Germany. Electronic mail address: firstname.lastname@example.org. This work was supported by a scholarship of the Hochschulsonderprogramm II/AUFE des Deutschen Akademischen Austauschdienstes (DAAD) and carried out while visiting Prof. Olof Widlund at the Courant Institute in New York during the academic year 1993/94 and Dr. Barry Smith at Argonne National Laboratory, Argonne, IL. Dieser Artikel ist Teil einer geplanten Dissertation an der Mathematisch-Naturwissenschaftlichen Fakult?at der Westf?alischen Wilhelms-Universit?at.