Numerical Analysis 4

This course focuses on the study of advanced numerical methods for partial differential equations that allow for the use of polyhedral meshes. The first part of the course is devoted to analytical 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 state-of-the-art numerical methods. In the third part, we develop applications of these methods related to the research activities currently underway at IMAG: fluid mechanics, solid mechanics, and flow in porous media.

Introduction to Advanced Numerical Methods for PDEs.

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

Recommended prerequisites: Numerical Analysis Courses 1, 2, and 3 from the first year of the master’s program

A sample lesson plan is as follows:

1) General Framework 

            * Reminders

            * Error analysis and Strang's third lemma

            * Polyhedral meshes and sequences of regular meshes

            * Function spaces

            * Basic tools (trace inequalities, inverses, etc.)

2) The Hybrid High-Order (HHO) Method for Poisson

            * Local construction

            * Stabilization

            * Discrete problem

            * Flux formulation

            * Error analysis using the energy norm and the L2 norm

3) The Mixed High-Order Method

            * Mixed formulation of the Poisson problem

            * Discrete problem

            * Hybridization and its connection to the HHO method

4) Variations on the HHO method and connections to other methods (one or more topics to be selected based on the course’s progress)

            * 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