Goal of the course is to introduce the concepts of numerical modelling for partial differential equations. We will mainly consider elliptic equations and numerical methods based on finite elements, classic and mixed formulations. Moreover, the course addressed in detail during lab hours algorithmic topics and leads the students to the complete implementation of the proposed methods on MATLAB
Expected learning outcomes
The course provides the bases to face the finite element discretization of partial derivatives equations for significant differential model problems. At the end of the course, the student should be able to choose stable finite element discretizations for such problems and to provide convergence estimates. Moreover, he/she should be able, starting from the provided material, to implement a numerical code to solve the problems on a computer
Lesson period: Second semester
(In case of multiple editions, please check the period, as it may vary)
1. Fundamentals of analysis (weak derivatives, Sobolev spaces, variational formulations). 2. The finite element method for elliptic problems (such as heat diffusion) in primal form, piecewise polynomial spaces, convergence properties. 3. The finite element method for problems in mixed form. General theory, stability and convergence. Finite elements for Stokes. Finite elements for the diffusion problem in mixed form.
Prerequisites for admission
To properly face the course, the Student should have a basic knowledge of Functional Analysis. Some knowledge regarding the Approximation Theory in functional spaces will be useful.
The course will be given by means of traditional lessons, using the blackboard. Furthermore, for the lab lessons the software MATLAB will be used to implement the studied methods.
- D. Boffi, F. Brezzi, M.Fortin, "Mixed Finite Elements and Applications", Springer Series in Computational Mathematics - S.Brenner, R.Scott, "The Mathematical Theory of Finite Element Methods", Texts in Applied Mathematics, Springer - A. Quarteroni "Modellistica numerica per problemi differenziali", Springer-Verlag Italia - A. Ern, J.-L.Guermond, "Theory and Practice of Finite Elements", Applied Mathematics Sciences Series, Springer
Assessment methods and Criteria
The final examination consists of two parts: an oral exam and a lab exam.
- In the oral exam, the student will be required to illustrate results presented during the course and will be required to solve problems regarding Galerkin methods in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them. -The lab exam consists in developing a project, which will be assigned in advance by the professor. The project will be presented by the student during the oral exam. The lab portion of the final examination serves to assess the capability of the student to put a problem of numerical approximation of PDEs into context, find a solution and to give a report on the results obtained.
The complete final examination is passed if both the parts (oral, lab) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.