Course offered by Prof. Patrick Farrell of the University of Oxford and Prof. Emílio Carlos Nelli Silva in the first period of the PPGEM.

The course is based on the use of FENICS software


Prof. Dr. Patrick Farrell

Prof. Dr. Emílio Carlos Nelli Silva (sala MS-06)

The aim of this course is to teach the concepts of a finite element computational simulation methodology that allows starting from the variational formulation of the problem to solve partial differential equations. In this way,  concepts will be explored on the finite element method, variational calculation, high order  linear system solution techniques and other concepts involved in the method. As a tool will be  used FENICS software (free license), which is a recent computational tool that follows the concept of modeling from the variational formulation.

Topics covered in the course:
1. Introduction to the finite element method (Implementation of the Poisson equation, Verifying code with the method of manufactured solutions)
2. Constructive solid geometry; Geometric singularities in the L-shaped domain
3. Adding a potential term; Varying the type of boundary conditions; Integrating over subdomains
4. The Newton-Kantorovich method for PDEs
5. Nonlinear radiation boundary conditions and Continuation for nonlinear problems
7. Solution of time-dependent PDEs (PDEs on manifolds)
8. Mixed variational problems: the Stokes equations
9. Nonlinear hyperelasticity
10. Variational inequalities with semismooth Newton methods
11. Eigenvalue problems
12. Oneshot methods for PDE-constrained optimisation problems
13. Parallelism in PDE solvers with MPI and Algebraic and geometric multigrid methods
15. Schur complement preconditioners: the Stokes equations
16. Bifurcation analysis of PDEs including Navier-Stokes equations
Evaluation Method  – Based on Project (10-page report)

Location: To be held at Class room A4 


1) Cornelius Lanczos, The Variational Principles of Mechanics, Fourth Edition, ISBN-13: 978-0486650678 , Dover Publications, 2004

2) Anders Logg, Kent-Andre Mardal, Garth N. Wells, Automated Solution of Differential Equations by the Finite Element Method – The FEniCS Book, ISBN 978-3-642-23098-1, DOI 10.1007/978-3-642-23099-8, The Fenics Project, 2011

3) J.N. Reddy, An Introduction to the Finite Element Method, 3RD EDITION, McGraw-Hill, Inc., ISBN13: 9780072466850, 2006

4) Susanne Brenner, Ridgway Scott, The Mathematical Theory of Finite Element Methods, Springer, ISBN 978-0-387-75934-0, 2008