turkmath.org

The talk is concerned with the convergence analysis of finite element methods for the approximate solution of a system of nonlinear elliptic partial differential equations that arise in models of chemically reacting viscous incompressible fluids. The shear-stress appearing in the model involves a power-law type nonlinearity, where, instead of being a fixed constant, the power law-exponent is a function of a spatially varying nonnegative concentration function, which, in turn, solves a nonlinear convection-diffusion equation. In order to prove the convergence of the sequence of finite element approximations to a solution of this coupled system of nonlinear PDEs, a uniform H\"{o}lder norm bound needs to be derived for the sequence of finite element approximations to the concentration in a setting, where the diffusion coefficient in the convection-diffusion equation satisfied by the concentration is merely an L^\infty function. This necessitates the development of a finite element counterpart of the De Giorgi--Nash--Moser theory. Motivated by an early paper by Aguilera and Caffarelli (1986) in the simpler setting of Laplace's equation, we derive such uniform H\"older norm bounds on the sequence of continuous piecewise linear finite element approximations to the concentration. We then use this to deduce the convergence of the sequence of finite element approximations to a solution of the coupled system of nonlinear PDEs under consideration.

The talk is based on joint work with Seungchan Ko and Petra Pustejovska, and recent results obtained in collaboration with Lars Diening and Toni Scharle.