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 realize here that it suffices to choose the stabilization tensor as the identity times the meshsize, and hence neither fixed-point arguments nor related parameters are needed anymore 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 123 J Sci Comput 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, are presented.

Mixed Discontinuous Galerkin Method for Quasi-Newtonian Stokes Flows

“…Many researchers aim at studying the efficient numerical methods for quasi-Newtonian flows and related problems, such as the conforming and nonconforming finite element method [4,6,17], the mixed finite element method [5,21,22], the dual-mixed finite element method [18,31], the discontinuous Galerkin (DG) method [13,16,26,27], the weak Galerkin method [43] and the virtual element method [14,25] and so on. Traditionally, the numerical methods are studied based on the velocity-pressure variational formulation, where the velocity and the pressure are the main unknowns [8,11].…”

“…However, the L 2 error estimates of strain rate and stress are not optimal. Then, the HDG scheme with an RT-like (Raviart-Thomas) elements was studied in [27], and optimal error analysis is established; in addition, a reliable and efficient residual-based a posteriori error estimator was derived. Note that these methods are based on pseudostress formulation, in which the pseudostress tensor is not symmetric.…”

Error analysis of an unfitted HDG method for a class of non-linear elliptic problems

“…However, in the presence of ferroelectric materials, the permeability is affected by the total magnetic field B-which is proportional to the gradient of u-and the coefficient then takes the form κ = κ(∇u), leading to a quasi-linear equation that requires the more detailed treatment that will be the subject of this article. Some theoretical studies of the HDG method applied to quasilinear problems have been pursued recently [6,7,8], however these efforts are limited to polygonal domains. Moreover, the first reference does not consider non-linearities of the form κ(∇u), while in [7,8] the authors analyzed an augmented HDG discretization for a strictly quasi-linear problem arising from a non-linear Stokes flow using an approach based on a nonlinear version of the Babuska-Brezzi theory.…”

“…Some theoretical studies of the HDG method applied to quasilinear problems have been pursued recently [6,7,8], however these efforts are limited to polygonal domains. Moreover, the first reference does not consider non-linearities of the form κ(∇u), while in [7,8] the authors analyzed an augmented HDG discretization for a strictly quasi-linear problem arising from a non-linear Stokes flow using an approach based on a nonlinear version of the Babuska-Brezzi theory. As we will show, our analysis will be valid for both quasi-linear and semi-linear problems, will not require an augmented formulation and the domain may be piecewise smooth.…”

Fully Computable a Posteriori Error Bounds for Hybridizable Discontinuous Galerkin Finite Element Approximations

“…In comparison, there are relatively few works on a posteriori error estimates for the hybridizable discontinuous Galerkin (HDG) methods [22]. The a posteriori estimates for HDG methods that are currently available in the literature [17,20,25,26,31,35] are all of residual type, in which reliability is shown up to a generic (unknown) constant. This means that, while the associated estimation may be suitable as local refinement indicators, they cannot provide a quantitative stopping criterion.…”