250075 VU Numerical Methods for PDE (2017S)

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. At the same time, finite element codes will be developed in the computer laboratory. The last part of this course, depending on the students' interests, might concern with either other applications (fluid mechanics, electromagnetics, elasticity), or non standard finite element methods (as discontinuous Galerkin methods), or domain decomposition techniques.