Numerical analysis of divergence-free discontinuous Galerkin methods for incompressible flow problems

Abstract

In the first major contribution of this thesis, we present analysis of two lowest-order hybridizable discontinuous Galerkin methods for the Stokes problem, while making only minimal regularity assumptions on the exact solution. The methods under consideration have previously been shown to produce H(div)-conforming and divergence-free approximate velocities. Using these properties, we derive a priori error estimates for the velocity that are independent of the pressure. These error estimates, which assume only H^{1+s}-regularity of the exact velocity fields for any s in [0, 1], are optimal in a discrete energy norm. Error estimates for the velocity and pressure in the L^2-norm are also derived in this minimal regularity setting. In the second major contribution of this thesis, we extend this analysis to the setting of the steady Navier--Stokes problem. We begin by proposing a new divergence-free discontinuous Galerkin method for the steady Navier--Stokes problem, and we show that the resultant discretized problem admits a unique solution under a smallness condition on the problem data. We then present an error analysis of the method in the minimal regularity setting, and we take special care to properly estimate the nonlinear terms arising from convection. We show that it is possible to derive optimal a priori error estimates for the velocity in a discrete energy norm. Our velocity error estimates are independent of the pressure, and require only H^{1+s}-regularity of the exact velocity fields for s in (0, 1] in the two-dimensional case and s in (1/2, 1] in the three-dimensional case.