## Mathematical Methods for Engineering

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

**Dates: 2nd part postponed /online >> 25 May - 05 June 2020**

The lessons will be delivered in **asynchronous** mode (an audio/video file will be made available at that time). The tutorials will be delivered in **synchronous **mode (the students will be required to join a zoom meeting at that time).

**Timetable online 25 May - 05 June 2020: **

Date |
Lessons |
Tutorials |

Monday 25/05 |
9:30 - 12:30 |
----- |

Tuesday 26/05 |
9:30 - 12:30 |
14:30 - 17:30 |

Wednesday 27/05 |
9:30 - 12:30 |
14:30 - 17:30 |

Thursday 28/05 |
9:30 - 12:30 |
14:30 - 17:30 |

Friday 29/05 |
9:30 - 12:30 |
14:30 - 17:30 |

Wednesday 03/06 |
9:30 - 12:30 |
14:30 - 17:30 |

Thursday 04/06 |
9:30 - 12:30 |
------ |

Friday 05/06 |
9:30 - 12:30 /14:30 - 16:30 |
------ |

EXAM: WEDNESDAY 10/06 |
14:30 - 17:30 |

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.