## Discrete De Giorgi-Nash-Moser Theory

### With Endre Suli (Oxford)

# Discrete De Giorgi-Nash-Moser Theory: Analysis and Applications

The talk is concerned with a class of numerical methods for the approximate solution of a system of nonlinear elliptic partial differential equations that arise in models of chemically-reacting viscous incompressible non-Newtonian fluids. In order to prove the convergence of the numerical method under consideration one needs to derive a uniform Hölder norm bound on the sequence of approximations in a setting where the diffusion coefficient in the convection-diffusion equation involved in the system is merely a bounded function with no additional regularity. This necessitates the development of a discrete counterpart of De Giorgi’s elliptic regularity theory, which is then used, in combination with various weak compactness techniques, to deduce the convergence of the sequence of numerical solutions to a weak solution of the system of partial differential equations. The theoretical result are illustrated by numerical experiments for a model of the synovial fluid, a non-Newtonian chemically-reacting incompressible fluid contained in the cavities of human joints.

- Speaker: Endre Suli (Oxford)
- Thursday 11 May 2023, 15:00–16:00
- Venue: Centre for Mathematical Sciences, MR14.
- Series: Applied and Computational Analysis; organiser: Matthew Colbrook.