250048 VO Numerics of Partial Differential Equations (2021W)

The course mainly focuses on Finite Element Methods for the numerical approximation of Partial Differential Equations. Three aspects of the finite element method will be considered: i) theoretical foundations, ii) examples of applications to the numerical approximation of partial differential equations arising in different application domains, iii) implementation details. After revising some basic concepts in functional analysis, finite element methods for the Poisson problem will be introduced; their stability and error analysis, as well as the basic tools for their implementation, will be presented. Then, finite element approximations of the heat equation, of the Helmholtz problem and of advection-dominated advection-diffusion problems will be considered, discussing the respective specific issues. Implementation details will be discussed. The last part of this course, depending on the students' interests, will concern either other applications (fluid mechanics, electromagnetics, elasticity), or non standard finite element methods (as discontinuous Galerkin methods).