Finite Element 1

– CODE : G5MSMEL1

– ECTS : 3

– VOLUME : CM 15h, TD 9h, TP 9h

– GRADE : (2 P1 + +3 P2 +TP) / 6

– CALENDAR : from mid-September 2026 to mid-November 2026

COURSE SUMMARY

The course focuses on the theoretical formulation of the finite element method and its implementation in the FreeFEM++ framework based on the weak formulation of a linear problem. It is designed for second-year Master’s students and assumes that the basic concepts of linear elasticity problems are known. The course combines theoretical foundations with practical programming exercises throughout.

The theoretical component of the course focuses on the mathematical foundations of the finite element method. It introduces the weak formulation of partial differential equations and the selection of the corresponding Sobolev function spaces. The course then examines the approximation of the solution in a finite-dimensional space using the Galerkin method. The practical construction of finite elements is subsequently presented, including shape functions, numerical integration and quadrature schemes, integration points, and the computation of element stiffness matrices. Finally, the assembly of the global system of equations is discussed. To consolidate these concepts, a Python-based practical exercise is carried out on a simple example encompassing all the key steps of the finite element formulation.

Throughout the course, practical programming work is performed to become familiar with the FreeFEM++ framework. Fundamentals of C++ programming are first recalled. Then, the specific functionalities provided by FreeFEM++, corresponding to the theoretical notions, are presented, including mesh generation, numerical integration, function space selection and weak formulation implementation. Display tools such as FreeFEM++ plotting tools and VTK visualization method, associated to the Paraview software, are presented. Practical simulations are then carried out to provide hands-on experience with the software.