After this course, the student is able to apply the finite element method to a given partial differential equation. In particular, the student is able to
- Apply the standard Galerkin approximation in the Lagrange Finite Element Space to a chosen problem;
- Assess and reflect the accuracy of the obtained solution using numerical error analysis;
- Apply a corresponding mixed finite element method and reflect its well-posedness;
- Assess the accuracy of the mixed finite element method using numerical error analysis;
- Modify the finite element method in order to obtain reliable and efficient approximations.
The finite element method (FEM) is the most widely used method for solving partial differential equations numerically. These equations arise in engineering and mathematical modelling. The range of the applications is immense, although typical applications include solid and fluid mechanics, electromagnetism or biology.
The finite element method is based on a variational equation, for which we will first develop some analytical insight.
We will then consider appropriate discrete subspaces (the finite element subspaces) in which we can solve the variational equation.
This leads to several algorithms for which the error analysis will be discussed.
A further focus will be the design of a corresponding error estimator and its use in an adaptive strategy.
This course has, therefore, two aims:
- Expose the student to the variational framework on which the finite element method is based and enrich the students' knowledge on numerical analysis.
- Practicing programming using Matlab with emphasis on efficiency and correctness.
After this course, the student has the ability to implement the finite element method for a given problem efficiently and reliably.
The course consists of lectures providing the analysis framework to the finite element methods. Tutorials will focus on the application and on the implementation.
During the tutorials, the student will work on a personal project, dealing with a given, possibly chosen, partial differential equation. The student will have the opportunity to present the milestones of his project. A typical project includes: the derivation of a first variational formulation and the reflection on its well-posedness, its modification to obtain a reliable and efficient formulation, numerical analysis of the modified formulation, numerical results.
A group project is only possible, if all students of the group have a distinct goal. A typical goal project includes an introduction to the problem, and different directions for each student of the group, for instance: reliable error estimator, comparison to a mixed formulation, extension to a non-linear problem.