## Mathematical methods for engineering

**Prof. Alberto Valli, Prof. Ana Alonso Rodriguez**

**Timetable 2013**

Feb. -March 2013 |
Hours |
Room |

18 February | 14:30-17:30 | D1 |

19 - 22 February | 9:30-12:30 / 14:30-15:30 | D1 |

25 February | 9:30-12:30 | D1 |

26 February | 9:30-12:30 | C2 |

27 February | 9:30-12:30 | R1 |

28 February | 9:30-12:30 | R1 |

1 March | 9:30-12:30 | D1 |

Tutorials (Alonso) | ||

21-22 February | 15:30-17:30 | PC-EST |

25 - 28 February | 14:30-17:30 | PC-EST |

Assessment: 1 March | 14:30-18:30 | PC-EST |

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**Programme**

**1. Separation of variables**

1.1. Heat equation in one space variable

1.2. Wave equation in one space variable

1.3. Complete orthonormal basis and related Fourier expansion

1.4. Sturm-Liouville problems for second order linear symmetric elliptic operators

**2. Fundamental solutions and Green functions**

2.1. Concentrated unit impulse

2.2. Fundamental solution of a linear operator L.

2.3. Fundamental solution of the Laplace operator in two and three variables

2.4. Green function in a bounded domain

**3. Integral equations and the boundary element method **

3.1. Singular integrals

3.2. Green formulae

3.3. Integral equation for the Dirichlet and Neumann boundary data

3.4. The boundary element method.

**4. Weak formulation and the finite element method**

4.1. Weak formulation of second order linear elliptic boundary value problems

4.2. Minimization problems in the calculus of variations

4.3. Lax-Milgram lemma and its consequences

4.4. Galerkin approximation method

4.5. The finite element method

4.6. Mixed formulation of second order linear elliptic equations, and the Stokes problem

4.7. Ladyzhenskaya-Babuska-Brezzi condition

4.8. Mixed finite element methods

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**Tutorials**

A minimum number of 15 hrs is foreseen for tutorials and exercises (prof. Alonso)

T.1. The boundary element method: remarks on programming

T.2. The finite element method, 1 (classical formulations): remarks on programming

T.3. The finite element method, 2 (mixed formulations): remarks on programming

T.4. Freefem: an example of finite element software.

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**Assessment**

The final test consists of a written exercise followed by an implementation issue. 4 ECTS will be assigned upon positive results in the assessment.

**References**

C.C. Mei, *Mathematical Analysis in Engineering*, Cambridge University Press, 1995 (Selected subjects from Chapters 2, 3 and 6). F. Paris and J. Cañas, *Boundary Element Method*, Oxford University Press, 1997 (Selected subjects from Chapters 1-3).

A. Quarteroni and A. Valli, *Numerical Approximation of Partial Differential Equations*, Springer 1997 (2nd printing) (Selected subjects from Chapters 3, 5-9).

IMPORTANT NOTICE: all external students, scholars and professionals interested in the course can participate upon payment of the fees as specified in the website of the __2013 Trento Winter School on Numerical Methods__.