A superconvergent HDG method for the incompressible Navier–Stokes equations on general polyhedral meshes

Qiu, Weifeng; Shi, Ke

doi:10.1093/imanum/drv067

Cited by 73 publications

(62 citation statements)

References 33 publications

(69 reference statements)

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“…For more general triangulations we refer to [27] where a new HDG method for the incompressible Navier-Stokes equations on general polyhedral meshes is proposed.…”

Section: The Hybridizable Discontinuous Galerkin Methodsmentioning

confidence: 99%

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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“…For more general triangulations we refer to [27] where a new HDG method for the incompressible Navier-Stokes equations on general polyhedral meshes is proposed.…”

Section: The Hybridizable Discontinuous Galerkin Methodsmentioning

confidence: 99%

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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“…Under assumptions (9) and (44), let u be the solution of problem (12) and u h be the solution of virtual problem (47).…”

Section: \Biggl( \Summentioning

confidence: 99%

“…We cite here the recent works [36,13,4,53,30,2,35] and [8,24,5], for instance. Finally, some examples of other numerical methods for the Stokes or Navier--Stokes equations that can handle polytopal meshes are [33,47,32].In this paper, we initiate the development of the VEM for the Navier--Stokes equations. We limit the study to two-dimensional domains and to diffusion dominated cases.…”

mentioning

confidence: 99%

Virtual Elements for the Navier--Stokes Problem on Polygonal Meshes

Veiga¹,

Lovadina²,

Vacca³

2018

SIAM J. Numer. Anal.

171

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Abstract. A family of virtual element methods for the two-dimensional Navier--Stokes equations is proposed and analyzed. The schemes provide a discrete velocity field which is pointwise divergence-free. A rigorous error analysis is developed, showing that the methods are stable and optimally convergent. Several numerical tests are presented, confirming the theoretical predictions. A comparison with some mixed finite elements is also performed.Key words. virtual element method, polygonal meshes, Navier--Stokes equations AMS subject classifications. 65N30, 76D05 DOI. 10.1137/17M11328111. Introduction. The virtual element method (VEM), introduced in [11,12], is a recent paradigm for the approximation of partial differential equation problems that shares the same variational background as the finite element methods. The original motivation of VEM is the need to construct an accurate conforming Galerkin scheme with the capability to deal with highly general polygonal/polyhedral meshes, including``hanging vertexes"" and nonconvex shapes. Among the Galerkin schemes, VEM is peculiar in that the discrete spaces consist of functions which are not known pointwise, but about which a limited set of information is available. This limited information is sufficient to construct the stiffness matrix and the right-hand side.The VEM has been developed for many problems; see, for example, [23,1,10,43,46,9,19,17,18,6,42,54,51,37,45]. More specifically, with regard to the Stokes problem, virtual elements have been developed in [3,28,15,25,26,52]. Moreover, VEM is also attracting growing interest for continuum mechanics problems within the engineering community. We cite here the recent works [36,13,4,53,30,2,35] and [8,24,5], for instance. Finally, some examples of other numerical methods for the Stokes or Navier--Stokes equations that can handle polytopal meshes are [33,47,32].In this paper, we initiate the development of the VEM for the Navier--Stokes equations. We limit the study to two-dimensional domains and to diffusion dominated cases. Although this is the simplest situation, it is nonetheless challenging and shows the satisfactory performance of the method; considering more complex cases will be a further step in future work. The presented scheme may be considered as a natural evolution of our recent divergence-free approach developed in [15] for the Stokes problem. However, the nonlinear convective term in the Navier--Stokes equations leads to the introduction of suitable projectors. These, in turn, suggest making use of an enhanced discrete velocity space [52], which is an improvement with respect

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“…This method was proposed by [19] and later analyzed by [21] for a single steady elliptic PDE, they obtained a superconvergent rate for the scalar variable for all k ≥ 0. This HDG method has been extended to study the PDEs with a convection term by [23,24]. However, the superconvergent rate was lost when k = 0.…”

Section: Introductionmentioning

confidence: 99%

A Superconvergent Ensemble HDG Method for Parameterized Convection Diffusion Equations

Chen¹,

Pi²,

Xu³

et al. 2019

SIAM J. Numer. Anal.

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We devised a first order time stepping ensemble hybridizable discontinuous Galerkin (HDG) method for a group of parameterized convection diffusion PDEs with different initial conditions, body forces, boundary conditions and coefficients in our earlier work [3]. We obtained an optimal convergence rate for the ensemble solutions in L ∞ (0, T ; L 2 (Ω)) on a simplex mesh; and obtained a superconvergent rate for the ensemble solutions in L 2 (0, T ; L 2 (Ω)) after an elementby-element postprocessing if polynomials degree k ≥ 1 and the coefficients of the PDEs are independent of time. In this work, we propose a new second order time stepping ensemble HDG method to revisit the problem. We obtain a superconvergent rate for the ensemble solutions in L ∞ (0, T ; L 2 (Ω)) without an element-by-element postprocessing for all polynomials degree k ≥ 0. Furthermore, our mesh can be any polyhedron, no need to be simplex; and the coefficients of the PDEs can dependent on time. Finally, we present numerical experiments to confirm our theoretical results.

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A superconvergent HDG method for the incompressible Navier–Stokes equations on general polyhedral meshes

Cited by 73 publications

References 33 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

Virtual Elements for the Navier--Stokes Problem on Polygonal Meshes

A Superconvergent Ensemble HDG Method for Parameterized Convection Diffusion Equations

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