Prof. Alberto Valli, Prof. Ana Alonso Rodriguez
|Feb. -March 2013||Hours||Room|
|19 - 22 February||9:30-12:30 / 14:30-15:30||D1|
|25 - 28 February||14:30-17:30||PC-EST|
|Assessment: 1 March||14:30-18:30||PC-EST|
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
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.
The final test consists of a written exercise followed by an implementation issue. 4 ECTS will be assigned upon positive results in the assessment.
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.