Mathematikgebäude, Raum M 614

In this lecture, I will consider the approximation of linear variational problems, symmetric and elliptic for simplicity. According to the classical Céa's lemma, the approximation of these problems by a conforming Ritz-Galerkin method is quasi-optimal, in the sense that the error of the method, measured in the energy norm, equals the best approximation error. In contrast, a simple argument reveals that many classical nonconforming methods (like the Crouzeix-Raviart or Discontinuou Galerkin methods) do not enjoy such property. Motivated by this observation, I will introduce a rather large class of possibly nonconforming methods and I will characterize the subclass of the quasi-optimal ones. For this purpose, I will propose notions of stability and consistency that are necessary and sufficient for quasi-optimality. The size of the constants involved in the analysis will also be discussed. To illustrate the abstract results, I will describe a modified version of the Crouzeix-Raviart method that is quasi-optimal for the Poisson problem. The lecture will cover a selection of arguments from my PhD thesis, which was written at the Università degli Studi di Milano, under the supervision of Prof. Andreas Veeser.

Dr. Pietro Zanotti

TU Dortmund