Numerical and theoretical study of the finite element method and its application

Lessons and computer lab practice

Oral examination.

Fundamental notions of Analysis and Numerical Analysis

Theory lessons will cover the following topics:
- fundamentals of Functional Analysis, with a particular emphasis on the W^{k,p} spaces and on primal variational formulations of elliptic problems
- approximation theory in Sobolev spaces: Deny-Lions Lemma and Brambe-Hilbert lemma
- Lagrange interpolation on n-simplices and corresponding interpolation error for Sobolev norms
- Galerkin method for elliptic problems and error estimates: Cea Lemma and duality techniques
- finite element methods for elliptic problems, with particular emphasis to the bidimensional case
- mixed formulation of elliptic problems and its Galerkin
discretization: existence, uniqueness, stability of the solution, and
error analysis.
- mixed finite elements for the Laplacian in mixed form (Darcy)
- finite elements for the Stokes system

Computer lab lessons will address the implementation of the finite
element method, in MATLAB language. In particular some of the following:
- data structure and algorithm for the triangulation of a planar region
- interpolation and numerical integration of funtions on the triangulation
- local matrices and assembling
- Dirichlet and Neumann boundary condition
- finite element method for the Poisson problem in primal form with P1 elements
- implementation of the RT element
- finite element method for the Poisson problem in mixed form (Darcy problem)

REMARK: This is a tentative program. Significant changes might occur,
also depending on the feedback provided by the Student during the
Lectures.

A. Quarteroni, R. Sacco, F. Saleri, P. Gervasio: "Matematica Numerica", Springer, 2014.
V. Comincioli: "Analisi Numerica, metodi, modelli, applicazioni", McGraw-Hill, 1995.
A. Quarteroni: "Numerical Models for Differential Problems", Springer, 2017.