NDLI: A Priori and a Posteriori Error Analyses of an Augmented HDG Method for a Class of Quasi-Newtonian Stokes Flows

Content Provider	Springer Nature Link
Author	Gatica, Gabriel N. Sequeira, Filánder A.
Copyright Year	2016
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 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.
Starting Page	1192
Ending Page	1250
Page Count	59
File Format	PDF
ISSN	08857474
Journal	Journal of Scientific Computing
Volume Number	69
Issue Number	3
e-ISSN	15737691
Language	English
Publisher	Springer US
Publisher Date	2016-06-14
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Nonlinear Stokes model Hybridized discontinuous Galerkin method Augmented formulation A posteriori error estimates Algorithms Computational Mathematics and Numerical Analysis ApplicationMathematics/Computational Methods of Engineering Theoretical, Mathematical and Computational Physics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Engineering Computational Theory and Mathematics Computational Mathematics Numerical Analysis Software

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