MATHEMATICAL METHODS

Theory of Partial differential equations

The diffusion equation. Well posed problems. Maximum principle. Fundamental solution. The global Cauchy problem.

The Laplace equation. Harmonic functions. Mean value properties and maximum principle. The fundamental solution and the Newtonian potential. The Green function.

The wave equation. The d'Alembert formula.

Elements of Functional Analysis. Normed and Banach Spaces. Lp spaces. Hilbert spaces. The Lax Milgram Theorem. Distributions. Sobolev spaces.

Variational formulation of elliptic problems.  Weak formulation af evolution problems. An application to control theory.

The Finite Element Method for partial differential equations

Basic Elements of numerical calculus: solution of linear syestems, approximation of functions and integrals, approximation of ODEs.

Finite element approximation of elliptic problems: variational formulation, algebraic formulation and error estimates.  Finite element approximation of parabolic equations: variational formulation, algebraic formulation and error estimates. Numerical solution of a model control problem.

Bibliography

S. Salsa, F. M. G: Vegni, A. Zaretti, P. Zunino, "A Primer on PDEs. Models, Methods, Simulations", Springer
S. Salsa, G. Verzini, "Equazioni a derivate parziali". Complementi ed esercizi, Springer.

A. Quarteroni, "Numerical Models for Differential Problems" , Editore: Springer http://link.springer.com/book/10.1007/978-88-470-0494-8/page/1

Grading Policy

There will be only a written exam.

e-book available