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### Finite Elements

Professors:
Sangalli Giancarlo, Boffi Daniele
Year:
2015/2016
Course code:
500679
ECTS:
9
SSD:
MAT/08
DM:
270/04
Lessons:
72
Period:
I semester
Language:
Italian

#### Objectives

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

#### Teaching methods

Lessons and computer lab practice

#### Examination

Oral examination.

#### Prerequisites

Fundamental notions of Analysis and Numerical Analysis

#### Syllabus

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.

#### Bibliography

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.

#### Modules

Professor:
Sangalli Giancarlo
Lessons:
48
ECTS:
6
SSD:
MAT/08

Professor:
Boffi Daniele
Lessons:
24
ECTS:
3
SSD:
MAT/08

Dipartimento di Matematica ''F. Casorati''

Università degli Studi di Pavia - Via Ferrata, 5 - 27100 Pavia
Tel +39.0382.985600 - Fax +39.0382.985602