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

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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.

This talk is part of the Applied and Computational Analysis series.

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