Mathematics

We start with a simple example of constrained minimization problem in R^2 which is solved using Lagrange multipliers, and we review the Kuhn-Tucker theory behind necessary and sufficient conditions for optima. This problem is a prototype of a saddle-point problem for the Lagrangian arising when minimizing a quadratic functional on an abstract Hilbert space X with a constraint arising in a product of X with another Hilbert space M. The existence and uniqueness of a solution requires that the inf-sup condition (BB or LBB) is satisfied. We give examples of finite dimensional spaces X and M which arise when discretizing Darcy and Stokes flow problems using simple mixed finite element approaches. In this talk we focus on the algebraic flavor of the inf-sup condition and on how to prevent the linear algebraic suicide which causes instabilities of the numerical solution; the analytical issues, convergence and stabilization will be discussed in later talks.