## Preprint 2015-32

## Gabriel N. Gatica, Filander A. Sequeira:

### A priori and a posteriori error analyses of an augmented HDG method for a class of quasi-Newtonian Stokes flows

### Abstract:

In a recent work we developed a new hybridizable discontinuous Galerkin (HDG) method for a class of nonlinear Stokes models arising in quasi-Newtonian fluids. The approach there uses the incompressibility condition to eliminate the pressure, sets the gradient of the velocity as an auxiliary unknown, and enriches the original formulation with convenient redundant equations, thus giving rise to an augmented scheme. However, the corresponding analysis, which makes use of a fixed point strategy that depends on a suitably chosen parameter, yields optimal rates of convergence for only two of the six resulting unknowns, whereas the reported numerical results, showing higher orders than predicted, support the conjecture that the a priori error estimates are not sharp. In the present paper, the main features of the aforementioned augmented formulation are maintained, but after introducing slight modifications of the finite element subspaces for the pseudostress and velocity, we are able to significantly improve our previous analyses and results. More precisely, on one hand we omit the utilization of any fixed-point argument and related parameters to establish the well-posedness of the discrete scheme, and on the other hand we now prove optimally convergent approximations for all the unknowns. Furthermore, we develop a reliable and efficient residual-based a posteriori error estimator, and propose the associated adaptive algorithm for our HDG approximation of the nonlinear model problem. Finally, several numerical results illustrating the performance of the method, confirming the theoretical properties of the estimator, and showing the expected behaviour of the adaptive refinements, even for an example not fully covered by the theory, are presented.

This preprint gave rise to the following definitive publication(s):

**Gabriel N. GATICA, Filander A. SEQUEIRA: ***A priori and a posteriori error analyses of an augmented HDG method for a class of quasi-Newtonian Stokes flows*. Journal of Scientific Computing, vol. 69, 3, pp. 1192-1250, (2016).