Numerical Methods for Partial Differential Equations 1

A.Y. 2019/2020
Overall hours
Learning objectives
Presentation of the finite element method for elliptic boundary value problems and analysis of the error of its approximate solution.
Expected learning outcomes
The understanding of the foundations of the finite element method. The ability to apply and implement the finite element method for stationary problems and to interpret the obtained numerical results.
Course syllabus and organization

Single session

Lesson period
First semester
Course syllabus
Introduction: 1-dimensional linear finite elements. Classical formulation of elliptic boundary values problems. Triangulations. Numerical integration on simplices. Finite element. Lagrange elements. Weak formulation and characterization of well-posed problems. Petrov-Galerkin methods. Sobolev spaces. Local and global approximation with piecewise polynomials. Convergence and a priori error bounds. Inverse estimates. Regularity of exact solutions. Quantities of interest. A posteriori error estimates.
Prerequisites for admission
Essential: Analysis, Linear Algebra, and experience in the programming language C.
Useful: Lebesgue integral, Numerical Linear Algrebra, and Constructive Approximation.
Teaching methods
Lecture, exercise and lab sessions.
Teaching Resources
·Dietrich Braess, Finite elements. Theory, fast solvers, and applications in solid mechanics, 3nd edition, Cambridge University Press, 2007
·S. C. Brenner, L. R. Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics 15, 2nd edition, Springer, 2002
·W. Hackbusch, Elliptic differential equations. Theory and numerical treatment, Springer Series in Computational Mathematics 18, Springer, 1987
·C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1987
·S. Larsson, V. Thomée, Partial differential equations with numerical methods, Texts in Applied Mathematics 45, Springer, 2000
·R. H. Nochetto, A. Veeser, Primer of Adaptive Finite Element Methods, in: Multiscale and Adaptivity: Modeling, Numerics and Applications, G. Naldi, G. Russo (ed.), Lecture Notes in Mathematics 2040, Springer, 2012.
·A. Quarteroni, Modellistica Numerica per Problemi Differenziali, Springer Italia, 2000
·A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations, Springer, 1991
·A. Schmidt, K. G. Siebert, Design of adaptive finite element software. The finite element toolbox ALBERTA, Lecture Notes in Computational Science and Engineering 42, Springer, 2005.
Assessment methods and Criteria
The final examination consists of two parts:
- the correction of a report summarizing a small project to be chosen and
- a final oral exam.

The project is chosen from a list that will be published at the end of the course, with specified validity. The project can be realized in collaboration with another person. The report summarizes the obtained results in at most 5 pages; it is recommended to write it not in collaboration. The correct submission of the project takes place at least two workdays before the oral exam and encompasses the report in pdf format, the used source codes (without executable files), and the name of the collaborator (if present).

The oral exam is on personal appointment after enrollment in an "appello". It starts with a discussion on the report. Subsequently, the student will be required to illustrate results presented during the course and will be required to solve problems in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

The complete examination is passed if the report and its discussion are evaluated positively and the oral exam is successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/08 - NUMERICAL ANALYSIS - University credits: 9
Laboratories: 36 hours
Lessons: 42 hours
on appointment by email
Skype or Microsoft Teams