Summary: PRECONDITIONING DISCRETE APPROXIMATIONS
OF THE REISSNERMINDLIN PLATE MODEL*
DOUGLAS N. ARNOLD, RICHARD S. FALK, and RAGNAR WINTHER§
Abstract. We consider iterative methods for the solution of the linear system of equations arising from
the mixed finite element discretization of the ReissnerMindlin plate model. We show how to construct a
symmetric positive definite block diagonal preconditioner such that the resulting linear system has spectral
condition number independent of both the mesh size h and the plate thickness t. We further discuss how
this preconditioner may be implemented and then apply it to efficiently solve this indefinite linear system.
Although the mixed formulation of the ReissnerMindlin problem has a saddle-point structure common
to other mixed variational problems, the presence of the small parameter t and the fact that the matrix
in the upper left corner of the partition is only positive semidefinite introduces new complications.
Key words. preconditioner, Reissner, Mindlin, plate, finite element
AMS(MOS) subject classifications (1991 revision). 65N30, 65N22, 65F10, 73V05
1. Introduction. The ReissnerMindlin plate model can be formulated as a saddle
point problem and discretized by mixed finite element methods. The resulting linear
algebraic system is symmetric and nonsingular, but indefinite. In this paper we show how
this system can be efficiently solved by preconditioned iterative methods. In particular,
we will establish bounds on the number of iterations necessary to achieve any desired
error reduction factor (in an appropriate norm) with the bounds independent of both the
discretization parameter h and the plate thickness t.