Discrete De Giorgi-Nash-Moser theory and the finite element approximation of chemically reacting fluids
A talk in the BI.discrete series by
Endre Süli from University of Oxford
| Abstract: | 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 Hölder
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 Hölder norm bounds on the sequence of continuous
piecewise
linear finite element approximations to the concentration. We then use
results
from the theory of variable-exponent Sobolev spaces equipped with a
Luxembourg
norm, Minty's method for monotone operators and an extension to
variable-exponent Sobolev spaces of the finite element version of the
Acerbi–Fusco Lipschitz-truncation method, originally developed in
classical
Sobolev spaces in collaboration with Lars Diening and Christian Kreuzer
(SIAM J. Numer. Anal. 51(2): 984–1015 (2014)),
to pass to the limit in the
coupled ystem 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. Meeting ID: 926 5310 0938 Passcode 1928 |