Analysis of an Augmented HDG Method for a Class of Quasi-Newtonian Stokes Flows

Gatica, Gabriel N.; Sequeira, Filánder A.

doi:10.1007/s10915-015-0008-5

Cited by 31 publications

(35 citation statements)

References 30 publications

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“…Indeed, the resulting augmented HDG scheme is reformulated as a fixed point problem, which, under the assumption that certain stabilization parameter is small enough, allows to apply a nonlinear version of the Babuška-Brezzi theory and the classical Banach fixed-point theorem. However, according to the numerical experiments reported in [21,Section 6], some of the unknowns show higher orders of convergence than predicted by the theoretical results, which suggests that the corresponding a priori error analysis is not sharp. In addition, those examples also showed that for large values of the stabilization parameter the method does not break down, thus insinuating that the restriction on the choice of this parameter could very well be just a technical assumption for the analysis.…”

Section: Introductionmentioning

confidence: 89%

“…In particular, a priori and a posteriori error analyses of nonaugmented and augmented mixed finite element methods for a quasi-Newtonian Stokes flow are developed in [20], whereas discontinuous Galerkin schemes are considered in [4] and more recently in [21]. More precisely, the application of the local discontinuous Galerkin (LDG) method to the aforementioned nonlinear problem was studied in [4] by using a pseudostressbased formulation in which the velocity, its gradient, and the pressure complete the set of unknowns.…”

Section: Introductionmentioning

confidence: 99%

“…More precisely, the application of the local discontinuous Galerkin (LDG) method to the aforementioned nonlinear problem was studied in [4] by using a pseudostressbased formulation in which the velocity, its gradient, and the pressure complete the set of unknowns. On the other hand, a hybridizable discontinuous Galerkin (HDG) method for the same model and approach from [4] is introduced and analyzed in [21]. This HDG formulation is enriched with two suitable redundant equations, thanks to which the well-posedness, that is the unique solvability, stability, and Céa's estimate of the proposed discrete scheme, can be easily established.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, those examples also showed that for large values of the stabilization parameter the method does not break down, thus insinuating that the restriction on the choice of this parameter could very well be just a technical assumption for the analysis. These drawbacks of the approach in [21] constitute the main motivation for the present contribution.…”

Section: Introductionmentioning

confidence: 99%

“…In the present paper, we reconsider the nonlinear model and the associated augmented HDG formulation from [21], and provide a significant improvement of the corresponding analyses and results. In fact, after suitable redefinitions of the finite element subspaces approximating the pseudostress and velocity, and without resorting to any fixed-point strategy, but only applying a nonlinear version of the well known Babuška-Brezzi theory, we are able to prove in a cleaner way the well-posedness of the discrete scheme and to establish optimal orders of convergence for all the unknowns.…”

Section: Introductionmentioning

confidence: 99%

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

Gatica

Sequeira

2016

J Sci Comput

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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.

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Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Gatica

Sequeira

2016

J Sci Comput

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Analysis of the HDG method for the stokes–darcy coupling

Gatica

Sequeira

2017

Numerical Methods Partial

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In this article, we introduce and analyze a hybridizable discontinuous Galerkin (HDG) method for numerically solving the coupling of fluid flow with porous media flow. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers‐Joseph‐Saffman law. We consider a fully‐mixed formulation in which the main unknowns in the fluid are given by the stress, the vorticity, the velocity, and the trace of the velocity, whereas the velocity, the pressure, and the trace of the pressure are the unknowns in the porous medium. In addition, a suitable enrichment of the finite dimensional subspace for the stress yields optimally convergent approximations for all unknowns, as well as a superconvergent approximation of the trace variables. To do that, similarly as in previous articles dealing with development of the a priori error estimates, we use the projection‐based error analysis to simplify the corresponding study. Finally, we provide several numerical results illustrating the good performance of the proposed scheme and confirming the optimal order of convergence provided by the HDG approximation. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 885–917, 2017

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The discrete duality finite volume method for a class of quasi‐Newtonian Stokes flows

Chen

et al. 2019

Numerical Methods Partial

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In this paper, we propose a discrete duality finite volume (DDFV) scheme for the incompressible quasi-Newtonian Stokes equation. The DDFV method is based on the use of discrete differential operators which satisfy some duality properties analogous to their continuous counterparts in a discrete sense. The DDFV method has a great ability to handle general geometries and meshes. In addition, every component of the velocity gradient can be reconstructed directly, which makes it suitable to deal with the nonlinear terms in the quasi-Newtonian Stokes equation. We prove that the proposed DDFV scheme is uniquely solvable and of first-order convergence in the discrete L 2 -norms for the velocity, the strain rate tensor, and the pressure, respectively. Ample numerical tests are provided to highlight the performance of the proposed DDFV scheme and to validate the theoretical error analysis, in particular on locally refined nonconforming and polygonal meshes. KEYWORDS discrete duality finite volume method, distorted, nonconforming and polygonal meshes, quasi-Newtonian Stokes equation 1

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Analysis of an Augmented HDG Method for a Class of Quasi-Newtonian Stokes Flows

Cited by 31 publications

References 30 publications

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

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

Analysis of the HDG method for the stokes–darcy coupling

The discrete duality finite volume method for a class of quasi‐Newtonian Stokes flows

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