**Lecturers: prof. Alberto Valli (Department of Mathematics); prof. Ana Alonso Rodriguez (Department of Mathematics)**

**Dates:** from Monday February 18th, 2019 to Friday March 1st, 2019

**Venue:** University of Trento, Department of Civil, Environmental and Mechanical Engineering (via Mesiano 77, I – 38123 Trento, Italy)

Contents:

**Theory**

- Partial differential equations (elliptic equations, parabolic equations, hyperbolic equations, boundary value problems).
- Separation of variables (solution of heat and wave equations by means of Fourier expansion, orthonormal bases, Sturm-Liouville problems, Bessel functions, Legendre and Chebyshev polynomials).
- Fundamental solutions and Green functions for elliptic equations (Dirac delta "function", distributions, fundamental solutions, Green functions, integral representation formula in terms of the Green function).
- Integral equations and the boundary element method for elliptic problems (Green formulae, interior and boundary integral representation formulae in terms of the fundamental solution, integral equation on the boundary, collocation and Galerkin formulations of the boundary element method, algebraic structure of the approximating problems).
- Weak formulation and the finite element method for elliptic problems (minimization problems, Euler equation of a functional, weak formulation, Lax-Milgram lemma, existence and uniqueness of the solution, Galerkin approximation, finite element methods and spectral methods, family of triangulations and basis functions, Céa lemma and error estimates, mixed formulation and Stokes problem, mixed finite element methods, Ladyzhenskaya-Babuska-Brezzi condition and error estimates, compatible choices of finite elements, algebraic structure of the discrete problems, other applications).

**Tutorials:**

- The boundary element method: remarks on programming.
- The finite element method, 1 (classical formulations) & programming
- The finite element method, 2 (mixed formulations) & programming
- FreeFEM: an example of finite element software.

__Lessons:__

__February 2019__

Monday 18, 14.30-17.30 - **room 1H**

Tuesday 19, 9.30-12.30 + 14.30-15.30 - **room 1H**

Wednesday 20, 9.30-12.30 + 14.30-15.30 - **room 1H**

Thursday 21, 9.30-13.30 -** room 1H**

Friday 22, 9.30-13.30 - **room 1H**

Monday 25, 9.30-12.30 -** room 1H**

Tuesday 26, 9.30-12.30 - **room 1H**

Wednesday 27, 9.30-12.30 - **room 1H**

Thursday 28, 9.30-12.30 -** room 1H**

__March 2019:__

Friday 1, 9.30-12.30 - **room 1H**

__Tutorials:__

Thursday 21, 15.30-17.30 - **room 1A (Mesiano)**

Friday 22, 15.30 - 17.30 - **room 1A (Mesiano)**

Monday 25, 14.30-17.30 - ** room 1A (Mesiano)**

Tuesday 26, 14.30-17.30 -** room A201 (Povo 1)**

Wednesday 27, 14.30-17.30 - **room A202 (Povo 1) **

Thursday 28, 14.30-17.30 - ** room A201 (Povo 1)**

**Exam: Friday March 1st, 14.30-17.30 ** **room 1A (Mesiano)**