Hilbert’s 19th problem and discrete De Giorgi-Nash-Moser theory: analysis and applications

This talk is concerned with the construction and mathematical analysis of a system of nonlinear partial differential equations featuring in a model of an incompressible non-Newtonian fluid, the synovial fluid, contained in the cavities of human joints. To prove the convergence of the numerical method one has to develop a discrete counterpart of the De Giorgi-Nash-Moser theorem, which guarantees a uniform bound on the sequence of continuous piecewise linear finite element approximations in a Hölder norm, for divergence-form uniformly elliptic partial differential equations with measurable coefficients.