The course is organized into 42 hours of frontal lesson and 36 hours of computer lab. Goal of the activity is to introduce the concepts of numerical modelling for partial differential equations. We will mainly consider elliptic equations. We will analyze 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
Fundamentals of analysis (weak derivatives, Sobolev spaces, variational formulations), the finite element method for elliptic problems (such as heat diffusion) in primal form, piecewise polynomial spaces, convergence properties. 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. Numerical methods based on finite elements for space/time parabolic problems. Example of a non-linear problem. The computer laboratory classes will be in the MATLAB Language.