An Optimal Preconditioner for a Class of Saddle Point Problems with a Penalty Term, Part Ii: General Theory

Iterative methods are considered for saddle point problems with a penalty term. A positive deenite preconditioner is constructed and it is proved that the condition number of the preconditioned system can be made independent of the discretization and the penalty parameters. Examples include the pure displacement problem in linear elasticity, the Timoshenko beam, and the Mindlin-Reissner plate. 1. Introduction. In this article, we extend the preconditioning strategy for saddle point problems with a penalty term, discussed in Klawonn 15], to a more general class of problems, including the Timoshenko beam and the Mindlin-Reissner plate. We consider problems of the form a(u; v) + b(v; p) = (f; v) 0 8v 2 V; b(u; q) ? t 2 c(p; q) = (g; q) 0 8q 2 M c t 2 0; 1]; where V; M and M c are Hilbert spaces with M c dense in M: In 15], we required that a(;) be V-elliptic and c(;) be equivalent to the L 2-inner product. In this paper, we prove that the condition number of the preconditioned system is bounded from above, independently of the discretization and the penalty parameters, if only the assumptions of the Babu ska-Brezzi theory hold, i.e. i) a(;) is elliptic on V 0 := fv 2 V : b(v; q) = 0 8q 2 Mg; ii) b(;) fulllls an inf-sup condition and iii) c(;) is non-negative. From these conditions, one can obtain an inf-sup and a sup-sup condition for the bilinear form A((u; p); (v; q)) := a(u; v) + b(v; p) + b(u; q) ? t 2 c(p; q) deened on the space X := V M c : The proof of the bound of the condition number is based mainly on the interpretation of these inf-sup and sup-sup conditions as providing an estimate for the condition number of A(;): Our preconditioning strategy can then be interpreted as introducing a new metric on V M c , i.e. performing a change of basis. Let us point out that a unifying multigrid approach for saddle point problems with a penalty term is developed in Brenner 8].