Numerical methods for partial differential equations
Scientific Disciplinary Sector (SSD)
MAT/08 - NUMERICAL ANALYSIS
Secondo semestre dal Mar 7, 2022 al Jun 10, 2022.
The course will discuss the theory and practice of Finite Element and Volume Methods. The theoretical part will follow course notes provided by the Instructor, advanced textbooks on Differential Equations, Iterative Methods for Sparse Linear Systems and numerical methods of Optimization. A part of the course will be held in a Laboratory setting where the methods discussed will be implemented in Matlab, using either the commercial version provided by Mathworks or else the open source version GNU Octave. In addition, high level scientific languages such as FreeFem++ and Clawpack for the numerical solution of elliptic, parabolic and hyperbolic equations will be introduced. At the end of the course the student is expected to have an excellent knowledge of the scientific and computational aspects of the techniques used to solve Partial Differential Equations by means of Finite Elements and Volumes.
Prerequisites: functional analysis, main methods for the numerical solution of differential equations.
The course will last 52 hours, 20 of which in the computer laboratory.
The course will discuss the following topics:
* Minimum Principle and the weak form, existence, uniqueness and regularity
* The Rayleigh-Ritz and Galerkin methods, finite element method, optimization methods, methods for the solution of sparse linear systems
* Transport and Diffusion equations, artificial diffusion, the generalized Galerkin method, discontinuous elements
* Hyperbolic and parabolic equations, finite volume method, semi and completely discretized problems
Visualizza la bibliografia con Leganto, strumento che il Sistema Bibliotecario mette a disposizione per recuperare i testi in programma d'esame in modo semplice e innovativo.
The purpose of the exam is to see if the student is able to recall and reproduce the theory and practice of Finite Elements. The exam will be oral. Alternatively, the student may choose to be examined on the basis of a specific software programming language. In this case, part of the evaluation will be replaced by a small project using the package FreeFem++ or Clawpack.