This course is meant to be complementary to the courses Numerical Solution of PDEs, Numerical Solution of PDEs Using the Finite Element Method, and Advanced FEM Techniques.
We discuss topics on advanced Finite Element Analysis, complemented with numerical implementations and examples provided via the deal.II C++ library (www.dealii.org) or the fenics library (www.fenicsproject.org).
The point of departure of the course is a generalization of the Lax Milgram Lemma, which is only a sufficient condition for wellposedness of elliptic problems in Hilbert spaces. We start by discussing necessary and sufficient conditions for wellposedness of linear problems posed in Banach spaces, and show the numerical implications of such results.
Such conditions are usually referred to as infsup conditions (Babuska 1972, Necas 1962). From the functional analysis point of view, the infsup conditions are a rephrasing of two fundamental theorems by Banach: the Closed Range Theorem and the Open Mapping Theorem, and are sometimes referred to as BNB (BanachNecasBabuska) conditions in the general Banach settings.
They become essentials when tackling saddle point problems, and in this settings their formalization is usually known as the Brezzi theory.
We will discuss in details well posedeness, stability, and convergence for saddle point problems, and provide some extension to nonconforming approximations.
A schematic of the syllabus follows:
 BanachNecasBabuska conditions (infsup conditions for linear problems in Banach spaces)
 Ladyzhenskaya–Babuška–Brezzi condition (infsup conditions for saddle point problems)
 Conforming numerical approximations of mixed problems (Stokes problem, Mixed Poisson (or Darcy problem))
 Nonconforming numerical approximations of mixed problems
 Applications and examples
References

Ern, A., Guermond, J.L., 2004. Theory and Practice of Finite Elements, Applied Mathematical Sciences. Springer New York, New York, NY. https://doi.org/10.1007/9781475743555
 Boffi, D., Brezzi, F., Fortin, M., 2013. Mixed Finite Element Methods and Applications, Springer Series in Computational Mathematics. Springer Berlin Heidelberg, Berlin, Heidelberg. https://doi.org/10.1007/9783642365195