ECTS
10 credits
Component
Faculty of Science
Description
This course covers advanced numerical methods for partial differential equations using polyhedral meshes. The first part of the course is devoted to analysis tools of general interest. The second part focuses on the design and analysis of Hybrid High-Order methods, an example of the latest generation of numerical methods. In the third part, applications of these methods are developed in relation to IMAG's research activities in fluid mechanics, solid mechanics and flows in porous media.
Objectives
Introduction to advanced numerical methods for PDEs.
Necessary prerequisites
Completion of a numerical analysis course at M1 level (Lagrange and mixed finite elements)
Recommended prerequisites: Numerical Analysis courses 1, 2 and 3 in the first year of the Master's program.
Syllabus
An indicative lesson plan is as follows:
1) General framework
* Reminders
* Error analysis and Strang's third lemma
* Polyhedral meshes and regular mesh sequences
* Function spaces
* Basic tools (trace inequalities, inverses, etc.)
2) Hybrid High-Order (HHO) method for Poisson
* Local construction
* Stabilization
* Discrete problem
* Flow formulation
* Energy standard and L2 standard error analysis
3) The Mixed High-Order method
* Mixed formulation of the Poisson problem
* Discrete problem
* Hybridization and link with the HHO method
4) Variations on the HHO method and links with other methods (choose one or more arguments depending on the progress of the course)
* The k-1, k and k+1 variants of the HHO method
* The Virtual Finite Element method
* The GDM method
5) Applications (depending on course progress)
* Diffusion-advection-reaction
* Linear elasticity
* Incompressible flows
(Reference "The Hybrid High-Order methods for Polytopal Meshes. Design, Analysis, and Applications" by D. A. Di Pietro and J. Droniou, Springer, 2020)
Further information
Hourly volumes :
CM :33
TD : 0
TP:0
Land: 0