ECTS
10 credits
Component
Faculty of Science
Description
This course deals with the study of advanced numerical methods for partial differential equations allowing the use of polyhedral meshes. The first part of the course is devoted to analysis tools of general interest. In the second part, we focus on the design and analysis of Hybrid High-Order methods, which are an example of the latest generation of numerical methods. In the third part, we develop applications of these methods in connection with the research activities present at IMAG: fluid mechanics, solid mechanics and flows in porous media.
Objectives
Introduction to advanced numerical methods for PDEs.
Necessary pre-requisites
Have taken a numerical analysis course at the M1 level (Lagrange and mixed finite elements)
Recommended prerequisites: Numerical Analysis courses 1,2 and 3 of the first year of the Master
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) The Hybrid High-Order (HHO) method for Poisson
* Local construction
* Stabilization
* Discrete problem
* Flow formulation
* Error analysis in energy standard and L2 standard
3) The Mixed High-Order method
* Mixed formulation of the Poisson problem
* Discrete problem
* Hybridization and link with the HHO method
4) Variations of the HHO method and links with other methods (one or more arguments to choose from 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 the progress of the course)
* 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)
Additional information
Hourly volumes:
CM :33
TD : 0
TP :0
Land : 0