December  2021, 29(6): 3629-3647. doi: 10.3934/era.2021054

A simple virtual element-based flux recovery on quadtree

Department of Mathematics and Statistics, Washington University in St. Louis, St. Louis, MO 63105, USA

* Corresponding author: Shuhao Cao

Received  November 2020 Revised  June 2021 Published  December 2021 Early access  July 2021

Fund Project: The first author is supported in part by NSF grants DMS-1913080 and DMS-2136075

In this paper, we introduce a simple local flux recovery for $ \mathcal{Q}_k $ finite element of a scalar coefficient diffusion equation on quadtree meshes, with no restriction on the irregularities of hanging nodes. The construction requires no specific ad hoc tweaking for hanging nodes on $ l $-irregular ($ l\geq 2 $) meshes thanks to the adoption of virtual element families. The rectangular elements with hanging nodes are treated as polygons as in the flux recovery context. An efficient a posteriori error estimator is then constructed based on the recovered flux, and its reliability is proved under common assumptions, both of which are further verified in numerics.

Citation: Shuhao Cao. A simple virtual element-based flux recovery on quadtree. Electronic Research Archive, 2021, 29 (6) : 3629-3647. doi: 10.3934/era.2021054
References:
[1]

R. AndersonJ. AndrejA. BarkerJ. BramwellJ.-S. CamierJ. C. V. DobrevY. DudouitA. FisherT. KolevW. PaznerM. StowellV. TomovI. AkkermanJ. DahmD. Medina and S. Zampini, MFEM: A modular finite element library, Computers & Mathematics with Applications, 81 (2021), 42-74.   Google Scholar

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Z. Cai and S. Cao, A recovery-based a posteriori error estimator for H(curl) interface problems, Comput. Methods in Appl. Mech. Eng., 296 (2015), 169-195.  doi: 10.1016/j.cma.2015.08.002.  Google Scholar

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Z. Cai and S. Zhang, Recovery-based error estimators for interface problems: conforming linear elements, SIAM J. Numer. Anal., 47 (2009), 2132-2156.  doi: 10.1137/080717407.  Google Scholar

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A. CangianiE. H. GeorgoulisT. Pryer and O. J. Sutton, A posteriori error estimates for the virtual element method, Numer. Math., 137 (2017), 857-893.  doi: 10.1007/s00211-017-0891-9.  Google Scholar

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S. Cao and L. Chen, Anisotropic error estimates of the linear nonconforming virtual element methods, SIAM J. Numer. Anal., 57 (2019), 1058-1081.  doi: 10.1137/18M1196455.  Google Scholar

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C. Carstensen and J. Hu, Hanging nodes in the unifying theory of a posteriori finite element error control, J. Comput. Math., 27 (2009), 215-236.   Google Scholar

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J. Červený, V. Dobrev and T. Kolev, Nonconforming mesh refinement for high-order finite elements, SIAM J. Sci. Comput., 41 (2019), C367-C392. doi: 10.1137/18M1193992.  Google Scholar

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L. Chen, iFEM: An Innovative Finite Element Methods Package in MATLAB, Technical report, 2008, URLhttps://github.com/lyc102/ifem. Google Scholar

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Z. Chen and S. Dai, On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients, SIAM J. Sci. Comput., 24 (2002), 443-462.  doi: 10.1137/S1064827501383713.  Google Scholar

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H. ChiL. Beirão da Veiga and G. H. Paulino, A simple and effective gradient recovery scheme and a posteriori error estimator for the virtual element method (VEM), Comput. Methods Appl. Mech. Engrg., 347 (2019), 21-58.  doi: 10.1016/j.cma.2018.08.014.  Google Scholar

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F. Dassi, J. Gedicke and L. Mascotto, Adaptive virtual element methods with equilibrated fluxes, arXiv preprint, arXiv: 2004.11220. Google Scholar

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L. DemkowiczJ. T. OdenW. Rachowicz and O. Hardy, Toward a universal hp adaptive finite element strategy, part 1. constrained approximation and data structure, Comput. Methods Appl. Mech. Engrg., 77 (1989), 79-112.  doi: 10.1016/0045-7825(89)90129-1.  Google Scholar

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P. Di StolfoA. SchröderN. Zander and S. Kollmannsberger, An easy treatment of hanging nodes in $hp$-finite elements, Finite Elem. Anal. Des., 121 (2016), 101-117.  doi: 10.1016/j.finel.2016.07.001.  Google Scholar

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A. Ern and M. Vohralík, Flux reconstruction and a posteriori error estimation for discontinuous Galerkin methods on general nonmatching grids, C. R. Math. Acad. Sci. Paris, 347 (2009), 441-444.  doi: 10.1016/j.crma.2009.01.017.  Google Scholar

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[24]

H. GuoC. Xie and R. Zhao, Superconvergent gradient recovery for virtual element methods, Math. Models Methods Appl. Sci., 29 (2019), 2007-2031.  doi: 10.1142/S0218202519500386.  Google Scholar

[25]

L. Mascotto, Ill-conditioning in the virtual element method: Stabilizations and bases, Numer. Methods Partial Differential Equations, 34 (2018), 1258-1281.  doi: 10.1002/num.22257.  Google Scholar

[26]

P. ŠolínJ. Červenỳ and I. Doležel, Arbitrary-level hanging nodes and automatic adaptivity in the $hp$-FEM, Math. Comput. Simulation, 77 (2008), 117-132.  doi: 10.1016/j.matcom.2007.02.011.  Google Scholar

[27]

R. Verfürth, Error estimates for some quasi-interpolation operators, M2AN Math. Model. Numer. Anal., 33 (1999), 695-713.  doi: 10.1051/m2an:1999158.  Google Scholar

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O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. part 1: The recovery technique, Internat. J. Numer. Methods Engrg., 33 (1992), 1331-1364.  doi: 10.1002/nme.1620330702.  Google Scholar

show all references

References:
[1]

R. AndersonJ. AndrejA. BarkerJ. BramwellJ.-S. CamierJ. C. V. DobrevY. DudouitA. FisherT. KolevW. PaznerM. StowellV. TomovI. AkkermanJ. DahmD. Medina and S. Zampini, MFEM: A modular finite element library, Computers & Mathematics with Applications, 81 (2021), 42-74.   Google Scholar

[2]

W. Bangerth, R. Hartmann and G. Kanschat, deal.II - a general purpose object oriented finite element library, ACM Trans. Math. Software, 33 (2007), Art. 24, 27 pp. doi: 10.1145/1268776.1268779.  Google Scholar

[3]

R. E. Bank and J. Xu, Asymptotically exact a posteriori error estimators, part ii: General unstructured grids, SIAM J. Numer. Anal., 41 (2003), 2313-2332.  doi: 10.1137/S0036142901398751.  Google Scholar

[4]

L. Beirão da VeigaF. BrezziA. CangianiG. ManziniL. D. Marini and A. Russo, Basic principles of virtual element methods, Mathematical Models and Methods in Applied Sciences, 23 (2013), 199-214.  doi: 10.1142/S0218202512500492.  Google Scholar

[5]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, Serendipity face and edge VEM spaces, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 28 (2017), 143-180.  doi: 10.4171/RLM/756.  Google Scholar

[6]

C. Bernardi and R. Verfürth, Adaptive finite element methods for elliptic equations with non-smooth coefficient, Numer. Math., 85 (2000), 579-608.  doi: 10.1007/PL00005393.  Google Scholar

[7]

S. Berrone and A. Borio, Orthogonal polynomials in badly shaped polygonal elements for the virtual element method, Finite Elem. Anal. Des., 129 (2017), 14-31.  doi: 10.1016/j.finel.2017.01.006.  Google Scholar

[8]

F. BrezziR. S. Falk and L. D. Marini, Basic principles of mixed virtual element methods, ESAIM Math. Model. Numer. Anal., 48 (2014), 1227-1240.  doi: 10.1051/m2an/2013138.  Google Scholar

[9]

R. Bruce Kellogg, On the Poisson equation with intersecting interfaces, Applicable Anal., 4 (1974), 101-129.  doi: 10.1080/00036817408839086.  Google Scholar

[10]

Z. Cai and S. Cao, A recovery-based a posteriori error estimator for H(curl) interface problems, Comput. Methods in Appl. Mech. Eng., 296 (2015), 169-195.  doi: 10.1016/j.cma.2015.08.002.  Google Scholar

[11]

Z. Cai and S. Zhang, Recovery-based error estimators for interface problems: conforming linear elements, SIAM J. Numer. Anal., 47 (2009), 2132-2156.  doi: 10.1137/080717407.  Google Scholar

[12]

A. CangianiE. H. GeorgoulisT. Pryer and O. J. Sutton, A posteriori error estimates for the virtual element method, Numer. Math., 137 (2017), 857-893.  doi: 10.1007/s00211-017-0891-9.  Google Scholar

[13]

S. Cao and L. Chen, Anisotropic error estimates of the linear nonconforming virtual element methods, SIAM J. Numer. Anal., 57 (2019), 1058-1081.  doi: 10.1137/18M1196455.  Google Scholar

[14]

C. Carstensen and J. Hu, Hanging nodes in the unifying theory of a posteriori finite element error control, J. Comput. Math., 27 (2009), 215-236.   Google Scholar

[15]

J. Červený, V. Dobrev and T. Kolev, Nonconforming mesh refinement for high-order finite elements, SIAM J. Sci. Comput., 41 (2019), C367-C392. doi: 10.1137/18M1193992.  Google Scholar

[16]

L. Chen, iFEM: An Innovative Finite Element Methods Package in MATLAB, Technical report, 2008, URLhttps://github.com/lyc102/ifem. Google Scholar

[17]

Z. Chen and S. Dai, On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients, SIAM J. Sci. Comput., 24 (2002), 443-462.  doi: 10.1137/S1064827501383713.  Google Scholar

[18]

H. ChiL. Beirão da Veiga and G. H. Paulino, A simple and effective gradient recovery scheme and a posteriori error estimator for the virtual element method (VEM), Comput. Methods Appl. Mech. Engrg., 347 (2019), 21-58.  doi: 10.1016/j.cma.2018.08.014.  Google Scholar

[19]

F. Dassi, J. Gedicke and L. Mascotto, Adaptive virtual element methods with equilibrated fluxes, arXiv preprint, arXiv: 2004.11220. Google Scholar

[20]

L. DemkowiczJ. T. OdenW. Rachowicz and O. Hardy, Toward a universal hp adaptive finite element strategy, part 1. constrained approximation and data structure, Comput. Methods Appl. Mech. Engrg., 77 (1989), 79-112.  doi: 10.1016/0045-7825(89)90129-1.  Google Scholar

[21]

P. Di StolfoA. SchröderN. Zander and S. Kollmannsberger, An easy treatment of hanging nodes in $hp$-finite elements, Finite Elem. Anal. Des., 121 (2016), 101-117.  doi: 10.1016/j.finel.2016.07.001.  Google Scholar

[22]

A. Ern and M. Vohralík, Flux reconstruction and a posteriori error estimation for discontinuous Galerkin methods on general nonmatching grids, C. R. Math. Acad. Sci. Paris, 347 (2009), 441-444.  doi: 10.1016/j.crma.2009.01.017.  Google Scholar

[23]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, 1986. doi: 10.1007/978-3-642-61623-5.  Google Scholar

[24]

H. GuoC. Xie and R. Zhao, Superconvergent gradient recovery for virtual element methods, Math. Models Methods Appl. Sci., 29 (2019), 2007-2031.  doi: 10.1142/S0218202519500386.  Google Scholar

[25]

L. Mascotto, Ill-conditioning in the virtual element method: Stabilizations and bases, Numer. Methods Partial Differential Equations, 34 (2018), 1258-1281.  doi: 10.1002/num.22257.  Google Scholar

[26]

P. ŠolínJ. Červenỳ and I. Doležel, Arbitrary-level hanging nodes and automatic adaptivity in the $hp$-FEM, Math. Comput. Simulation, 77 (2008), 117-132.  doi: 10.1016/j.matcom.2007.02.011.  Google Scholar

[27]

R. Verfürth, Error estimates for some quasi-interpolation operators, M2AN Math. Model. Numer. Anal., 33 (1999), 695-713.  doi: 10.1051/m2an:1999158.  Google Scholar

[28]

O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. part 1: The recovery technique, Internat. J. Numer. Methods Engrg., 33 (1992), 1331-1364.  doi: 10.1002/nme.1620330702.  Google Scholar

Figure 1.  For the upper right element $ K\in \mathcal{T} $, $ \mathcal{N}_K = \{z_2, z_4, z_5, z_6\} $. For $ K\in \mathcal{T}_{\mathrm{poly}} $, $ \mathcal{N}_K = \{z_i\}_{i = 1}^7 $
Figure 2.  The result of the L-shape example. (a) The adaptively refined mesh with 1014 DoFs. (b) Convergence in Example 1
Figure 3.  The result of the circular wave front example. (a) $ u_{\mathcal{T}} $ on a 3-irregular mesh with $ \# \mathrm{DoFs} = 1996 $, the relative error is $ 14.3\% $. (b) Convergence in Example 2
Figure 4.  Comparison of the adaptively refined meshes. (a) 1-irregular mesh, $ \# \mathrm{DoFs} = 1083 $, the relative error is $ 21.8\% $. (b) 4-irregular mesh, and $ \# \mathrm{DoFs} = 1000 $, the relative error is $ 17.8\% $
17,Section 4]). (b) The finite element approximation with $ \# \mathrm{DoFs} = 1736 $">Figure 5.  The result of the Kellogg example. (a) The adaptively refined mesh with $ \# \mathrm{DoFs} = 2001 $ on which the energy error is $ 0.0753 $, this number is roughly $ 75\% $ of the number of DoFs needed to achieve the same accuracy if using conforming linear finite element on triangular grid (see [17,Section 4]). (b) The finite element approximation with $ \# \mathrm{DoFs} = 1736 $
Figure 6.  The convergence result of the Kellogg example
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