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

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