Numerical Analysis 4

This course focuses on the study of advanced numerical methods for partial differential equations that allow the use of polyhedral meshes. The first part of the course is devoted to general analysis tools. The second part focuses on the design and analysis of Hybrid High-Order methods, which are an example of the latest generation of numerical methods. The third part develops applications of these methods in connection with the research activities carried out at IMAG: fluid mechanics, solid mechanics, and flows in porous media.

Introduction to advanced numerical methods for EDPs.

Have completed a numerical analysis course at the M1 level (Lagrange and mixed finite elements)

Recommended prerequisites: Numerical Analysis 1, 2, and 3 courses from the first year of the Master's program

A suggested 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 Poisson's problem

            * Discrete problem

            * Hybridization and link with the HHO method

4) Variations of the HHO method and links to other methods (one or more topics to be chosen 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)

Hourly volumes:

            CM: 33

            TD: 0

            TP: 0

            Land: 0