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. Some example of Finite Elements for the diffusion problem in mixed form
- elasticity problem and its FEM discretization: the volumetric locking phenomenon and some possible cures

Computer Lab lessons will address the implementation of the finite element method, in MATLAB language. In particular:
- 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, A. Valli: "Numerical Approximation of Partial Differential Equations", Springer-Verlag, 1994.


Braess, Dietrich. Finite elements: Theory, fast solvers, and applications in solid mechanics. Cambridge University Press, 2001.

Daniele Boffi, Franco Brezzi, and Michel Fortin. Mixed finite element methods and applications. Berlin: Springer, 2013.