## Advanced Finite Element Analysis

An advanced course dedicate to the analysis of finite element methods, as found in modern numerical analysis literature. A basic knowledge of Sobolev spaces is expected Detailed programA priori estimates

- Lax Milgram Lemma
- Cea’s Lemma
- Bramble Hilbert Lemma
- Inverse estimates
- Trace estimates

Stabilization mechanisms

## Numerical Solution of PDEs Using the Finite Element Method

**The Finite Element Method Using deal.II**

This is an intensive course that teaches how to use the finite element library deal.II (www.dealii.org).

Prerequisites: you should be familiar with C/C++, and with the Unix command line.

## Reduced Order Methods for Computational Mechanics

In this course we present reduced basis (RB) approximation and associated a posteriori error estimation for rapid and reliable solution of parametrized partial differential equations (PDEs). The focus is on rapidly convergent Galerkin approximations on a subspace spanned by "snapshots'"; rigorous and sharp a posteriori error estimators for the outputs/quantities of interest; efficient selection of quasi-optimal samples in general parameter domains; and Offline-Online computational procedures for rapid calculation in the many-query and real-time contexts.

## Topics in Computational Fluid Dynamics

- Introduction to CFD, examples.
- Constitutive laws
- Incompressible flows.
- Numerical methods for potential and thermal flows
- Boundary layer theory
- Thermodynamics effects, energy equation, enthalpy and entropy
- Vorticity equations
- Introduction to turbulence
- Numerical methods for viscous flows: steady Stokes equations
- Stabilisation (SUPG) and inf-sup condition