Learning objectives
- Knowledge and understanding of elementary concepts for the numerical modeling of elliptic and parabolic partial di_erential equations, in particular, based on finite difference, finite element, spectral and boundary element methods. - Ability to program the discussed numerical methods in Matlab for classical elliptic and parabolic linear equations, as well as the evaluation of algorithmic aspects, accuracy, stability and efficiency. - Autonomy of judgment in evaluating the approximation algorithms and the obtained results also through discussion with one's peers in possible team work. - Ability to clearly communicate the acquired concepts and to discuss the obtained results. - Ability to learn the drawbacks and the advantages of models and methods of resolution and to apply them in di_erent working and scientific contexts.
Prerequisites
- Basic methods and algorithms of numerical analysis.
- Knowledge of a programming language
Course unit content
Relevant background in analysis: Sobolev spaces, variational formulations of elliptic PDEs, relevant functional analysis - Finite dfference methods for elliptic problems: introduction, implementation, basic analysis. - Galerkin methods for elliptic problems: stability, error analysis, implementation of standard finite element methods. - Spectral methods for elliptic problems: spectral Galerkin and collocation methods. - Methods for parabolic problems: time discretization, implicit and explicit Euler method.
- Advanced topics, including boundary element methods and adaptive methods.
Full programme
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Bibliography
- "Finite Elements", D. Braess, Cambridge University Press, 2010. - "Numerical Approximation of Partial Di_erential Equations", A. Quarteroni, A. Valli, ed. Springer, 1994. - "Spectral Methods: Algorithms, Analysis and Applications", J. Shen, T. Tang, L.-L. Wang, Springer, 2011. -" Afinite element primer", D. J. Silvester, https://personalpages.manchester.ac.uk/sta_/david.silvester/primer.pdf - Sppectral Methods in Matlab", L. N. Trefethen, SIAM.
Teaching methods
During the lectures the contents of the course will be analyzed, highlighting the difficulties related to the introduced numerical techniques. Moreover, the course will consist of a part of autonomous re-elaboration, supervised by the professor, consisting in the application of the numerical techniques through laboratory programming. This activity will allow students to acquire the ability to deal with "numerical" difficulties and to evaluate the reliability and consistency of the obtained results
Assessment methods and criteria
The exam includes: - the assignment of a work for the application of numerical techniques introduced to solve a specific problem. The analysis of the results obtained by the student will allow to evaluate the acquisition of the above listed skills. In particular the threshold of sufficiency is fixed to the ability to achieve reliable numerical results. - an assessment of the knowledge through a discussion of topics of the course. The threshold of sufficiency consists in the knowledge of the discriminating characteristics of the various methods presented in the course.
Other information
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2030 agenda goals for sustainable development
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