Numerical Methods for Partial Differential Equations 1

A.Y. 2020/2021
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
Assessment methods and critera for its result
The assessment methods and critera for its results will not been changed, except for the fact that the final oral exam will be in person or on Microsoft Teams according to the existing directives in force at the time of the exam.

Course structure
The course will employ the platform Ariel for announcements and providing material. Lectures, exercises and lab sessions will be organized on Microsoft Teams (or Zoom) and can be followed according to the timetable or by means of recordings that will be available on Microsoft Teams (or Ariel).

The course syllabus will not be changed except for the fact that, if somebody does not succeed in installing ALBERTA, an alternative with the help of Matlab will be offered.
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 and quasi-optimality. 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.

Model implementation of 1-dimensional finite elements. Implementation of boundary value problems with the help of thie finite element toolbox ALBERTA.
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
Lectures, exercises 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 examination consists of two parts:
- the evaluation of a small project to be chosen and
- a final oral exam on personal appointment after enrollment in an "appello".

The project has to be chosen from a list that will be published at the the beginning of each exam session. The project can be realized in collaboration with another person; each member of the group has to complete its exam within the validity of the given project list. The correct submission of the project consists in a zip archive containing source codes (but no exectuable files for the antivirus) and a pdf report which summarizes the obtained results on at most 5 pages; it is recommended to write the report not in collaboration. The zip archive, together with the name of the collaborator (if present), has to be sent by email two workdays before the oral exam.

In order to arrange the date of the oral exam, the student has to be enrolled in the current "appello"; it is recommended to contact the professor at least one week before the desired date. Usually, the oral exam starts with a brief discussion on the report and lasts 45 minutes. The student is invited to present a copy of its report and to prepare for questions both concerning or not the chosen project. The exam cannot be repeated with the same project.

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 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