Adaptive variational multiscale methods for incompressible flow based on two local Gauss integrations

Zheng, Haibiao; Hou, Yanren; Shi, Feng

doi:10.1016/j.jcp.2010.05.038

Cited by 35 publications

(14 citation statements)

References 34 publications

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“…Following the principle of the uniform distribution of errors over each triangular element, the sizing function of the current mesh is modified with the method proposed in [8,24], where the readers can refer to the details. i at the refinement level l, which is calculated by…”

Section: Mesh Sizing Modificationmentioning

confidence: 99%

A modified bubble placement method and its application in solving elliptic problem with discontinuous coefficients adaptively

Zhou

Nie

Zhang

2016

International Journal of Computer Mathematics

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To solve the elliptic problem with discontinuous coefficients adaptively and efficiently which has strong singularity, one of the key steps is to generate mesh adaptively near region with discontinuous coefficients according to an a posteriori error estimator. For bubble placement method, the previous interaction force function used to obtain a high-quality nodes set is only determined by the ratio ω of the real and desired distance between bubbles, which is not suitable to generate mesh with a very small size. In this study, the force function is modified and it is determined both by the ratio ω and the sizes of bubbles. Meanwhile, in consideration of the quality of nodes and the computational efficiency, the equation of motion used to calculate the positions of bubbles is also modified. Moreover, a constrained bubble-type local mesh generation (BLMG) method is developed. The modified node placement method and constrained BLMG strategy are applied to solve elliptic problem with discontinuous coefficients adaptively. Several numerical experiments are reported to demonstrate that the adaptive method can be applied to various kinds of complicated geometries, and the triangles remain very well shaped at any particular refinement levels, even though the mesh size varies by several orders of magnitude. ARTICLE HISTORY

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Section: Mesh Sizing Modificationmentioning

confidence: 99%

A modified bubble placement method and its application in solving elliptic problem with discontinuous coefficients adaptively

Zhou

Nie

Zhang

2016

International Journal of Computer Mathematics

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“…Among them, the pressure projection method based on two local Gauss integrals is a preferable choice in that it is free of stabilization parameters, does not require any calculation of high-order derivatives or edges-based data structures, and can be implemented at an element level. Recent studies have been focused on stability and convergence of this method for the Stokes and Naiver-Stokes equations [15,16,17,35,36].…”

Section: Introductionmentioning

confidence: 99%

A stabilized finite element method based on two local Gauss integrations for a coupled Stokes–Darcy problem

Chen

2016

Journal of Computational and Applied Mathematics

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Please cite this article as: R. Li, J. Li, Z. Chen, Y. Gao, A stabilized finite element method based on two local Gauss integrations for a coupled Stokes-Darcy problem, Journal of Computational and Applied Mathematics (2015), http://dx. AbstractIn this paper, a stabilized mixed finite element method for a coupled steady Stokes-Darcy problem is proposed and investigated. This method is based on two local Gauss integrals for the Stokes equations. Its originality is to use a difference between a consistent mass matrix and an under-integrated mass matrix for the pressure variable of the coupled Stokes-Darcy problem by using the lowest equal-order finite element triples. This new method has several attractive computational features: parameter free, flexible, and altering the difficulties inherited in the original equations. Stability and error estimates of optimal order are obtained by using the lowest equal-order finite element triples (P 1 − P 1 − P 1 ) and (Q 1 − Q 1 − Q 1 ) for approximations of the velocity, pressure, and hydraulic head. Finally, a series of numerical experiments are given to show that this method has good stability and accuracy for the coupled problem with the Beavers-Joseph-Saffman-Jones and Beavers-Joseph interface conditions.

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“…There are many literatures on stabilized finite element methods for NavierStokes equations. Among them, we list some methods as follows: recently developed stabilized methods, such as, Galerkin least square method introduced in [4][5][6] by Franca, Hughes, and coworkers, and applied to some advective-diffusive models; residual-free bubbles (RFB) method [7][8][9], in which, the enrichment of the discrete finite element space by RFB; classical large eddy simulation (LES) approach in [10,11] which treats the large scales as an average in space given by convolution with an appropriate filter function; variational multiscale (VMS) methods, see for example, Hughes et al [12][13][14], they first reported VMS methods, Guermond [15] developed the subgrid modeling that is a variant of VMS methods, and Layton [16] discussed the connection between subgrid scale eddy viscosity and mixed methods, John, Kaya, and coworkers [17][18][19] applied the VMS methods to the Navier-Stokes equations and gave the theoretical error analysis, Zheng [20] improved the VMS method for the Navier-Stokes equations based on two local Gauss integrations, and other literatures on VMS methods [21][22][23][24]; two-level stabilization scheme [25] and local projection stabilization [26], both can be interpreted as a VMS method; and so on.…”

Section: Introductionmentioning

confidence: 99%

“…Based on the observation that this method is equivalent with, but save a lot of computational CPU time than the VMS method for the Taylor-Hood elements in the case of selecting lower order appropriate space for the velocity deformation tensor on the same mesh, Zheng et al provided a one-level method for the Navier-Stokes equations [20]. They also extend their work by combing with adaptive strategy [22].…”

Section: Introductionmentioning

confidence: 99%

A two‐level variational multiscale method for incompressible flows based on two local Gauss integrations

Li¹,

Mei²,

Li³

et al. 2013

Numerical Methods Partial

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In this article, a two-level variational multiscale method for incompressible flows based on two local Gauss integrations is presented. We solve the Navier-Stokes problem on a coarse mesh using finite element variational multiscale method based on two local Gauss integrations, then seek a fine grid solution by solving a linearized problem on a fine grid. In computation, we use the two local Gauss integrations to replace the projection operator without adding any variables. Stability analysis is performed, and error estimates of the method are derived. Finally, a series of numerical experiments are also given, which confirm the theoretical analysis and demonstrate the efficiency of the new method.

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Adaptive variational multiscale methods for incompressible flow based on two local Gauss integrations

Cited by 35 publications

References 34 publications

A modified bubble placement method and its application in solving elliptic problem with discontinuous coefficients adaptively

A modified bubble placement method and its application in solving elliptic problem with discontinuous coefficients adaptively

A stabilized finite element method based on two local Gauss integrations for a coupled Stokes–Darcy problem

A two‐level variational multiscale method for incompressible flows based on two local Gauss integrations

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