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Mathematical Methods for Engineering

Lecturer: Alberto Valli (UNITN/Dept. of Mathematics), Ana Maria Alonso Rodriguez (UNITN/Dept. of Mathematics)

Timetable: end of February - March 2024

Schedule: will be posted as soon as available

Programme:

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.

Detailed timetable:

February 27, 9:30-12:30 Room 1H

February 28, 9:30-12:30 Room 1H

February 29, 9:30-12:30 Room 1H

March 1, 9:30-12:30 Room 1H

March 4, 9:30-12:30 Room 1H

March 5, 9:30-12:30 Room 1H

March 6, 9:30-12:30 Room 1H

March 7, 9:00-12:00 (Room 1H) and tutorials 13:30 - 16:30  Room A107 (Povo) 

March 12, 9:30-12:30 (Room 1L) 

March 13, 9:30-12:30 (Room 1L) 

March 14, tutorials 13:30 - 16:30 Room A216 (Povo)

March 15, tutorials 8:30 - 11:30 Room A213 (Povo)

March 18, tutorials 13:30 - 16:30 Room A216 (Povo)

March 19, 9:30-11:30 (Room 1H) and tutorials 13:30 - 15:30 Room A218 (Povo)

March 20, 9:30-11:30 (Room 1H) and tutorials 13:30 - 15:30 Room A219 (Povo)

March 22, exam 14:00 - 17:00 Room A218 (Povo)

Duration: 50 hours (6 credits)

Registration: in order to register for the course, please send an email to dicamphd [at] unitn.it