April  2022, 9(2): 207-238. doi: 10.3934/jcd.2021020

A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem

1. 

T-5 Applied Mathematics and Plasma Physics Group, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

2. 

Dipartimento di Ingegneria Civile, Edile e Ambientale - ICEA, Università di Padova, 35131 Padova, Italy

* Corresponding author: G. Manzini

Received  March 2021 Revised  July 2021 Published  April 2022 Early access  December 2021

Fund Project: Dr. G. Manzini was supported by the LDRD-ER program of Los Alamos National Laboratory under project number 20180428ER. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001)

The Virtual Element Method (VEM) is a Galerkin approximation method that extends the Finite Element Method (FEM) to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element method for the numerical approximation of the Stokes problem to polygonal meshes in the framework of the virtual element method. In particular, we consider a straightforward application of the virtual element approximation space for scalar elliptic problems to the vector case and approximate the pressure variable through discontinuous polynomials. We assess the effectiveness of the numerical approximation by investigating the convergence on a manufactured solution problem and a set of representative polygonal meshes. We numerically show that this formulation is convergent with optimal convergence rates except for the lowest-order case on triangular meshes, where the method coincides with the $ {\mathbb{P}}_{{1}}-{\mathbb{P}}_{{0}} $ Scott-Vogelius scheme, and on square meshes, which are situations that are well-known to be unstable.

Citation: Gianmarco Manzini, Annamaria Mazzia. A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem. Journal of Computational Dynamics, 2022, 9 (2) : 207-238. doi: 10.3934/jcd.2021020
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003.

[2]

B. AhmadA. AlsaediF. BrezziL. D. Marini and A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl., 66 (2013), 376-391.  doi: 10.1016/j.camwa.2013.05.015.

[3]

P. F. AntoniettiL. Beirão da VeigaD. Mora and M. Verani, A stream virtual element formulation of the Stokes problem on polygonal meshes, SIAM J. Numer. Anal., 52 (2014), 386-404.  doi: 10.1137/13091141X.

[4]

P. F. AntoniettiG. Manzini and M. Verani, The fully nonconforming Virtual Element method for biharmonic problems, Math. Models Methods Appl. Sci., 28 (2018), 387-407.  doi: 10.1142/S0218202518500100.

[5]

P. F. AntoniettiG. Manzini and M. Verani, The conforming virtual element method for polyharmonic problems, Comput. Math. Appl., 79 (2020), 2021-2034.  doi: 10.1016/j.camwa.2019.09.022.

[6]

B. Ayuso de DiosK. Lipnikov and G. Manzini, The non-conforming virtual element method, ESAIM Math. Model. Numer. Anal., 50 (2016), 879-904.  doi: 10.1051/m2an/2015090.

[7]

B. Bang and D. Lukkassen, Application of homogenization theory related to Stokes flow in porous media, Appl. Math., 44 (1999), 309-319.  doi: 10.1023/A:1023084614058.

[8]

L. Beirão da VeigaF. BrezziA. CangianiG. ManziniL. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23 (2013), 199-214.  doi: 10.1142/S0218202512500492.

[9]

L. Beirão da VeigaF. Brezzi and L. D. Marini, Virtual elements for linear elasticity problems, SIAM J. Numer. Anal., 51 (2013), 794-812.  doi: 10.1137/120874746.

[10]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, The hitchhiker's guide to the virtual element method, Math. Models Methods Appl. Sci, 24 (2014), 1541-1573.  doi: 10.1142/S021820251440003X.

[11]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, Virtual element methods for general second order elliptic problems on polygonal meshes, Math. Models Methods Appl. Sci., 26 (2016), 729-750.  doi: 10.1142/S0218202516500160.

[12]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, H(div) and H(curl)-conforming VEM, Numer. Math., 133 (2016), 303-332.  doi: 10.1007/s00211-015-0746-1.

[13]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, Mixed virtual element methods for general second order elliptic problems on polygonal meshes, ESAIM Math. Model. Numer. Anal., 50 (2016), 727-747.  doi: 10.1051/m2an/2015067.

[14]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, Serendipity nodal VEM spaces, Comput. Fluids, 141 (2016), 2-12.  doi: 10.1016/j.compfluid.2016.02.015.

[15]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, Virtual element methods for general second order elliptic problems on polygonal meshes, Math. Models Methods Appl. Sci., 26 (2016), 729-750.  doi: 10.1142/S0218202516500160.

[16]

L. Beirão da VeigaA. ChernovL. Mascotto and A. Russo, Basic principles of hp virtual elements on quasiuniform meshes, Math. Models Methods Appl. Sci., 26 (2016), 1567-1598.  doi: 10.1142/S021820251650038X.

[17]

L. Beirão da VeigaF. Dassi and G. Vacca, The Stokes complex for virtual elements in three dimensions, Math. Models Methods Appl. Sci., 30 (2020), 477-512.  doi: 10.1142/S0218202520500128.

[18]

L. Beirão da VeigaV. GyryaK. Lipnikov and G. Manzini, Mimetic finite difference method for the Stokes problem on polygonal meshes, J. Comput. Phys., 228 (2009), 7215-7232.  doi: 10.1016/j.jcp.2009.06.034.

[19]

L. Beirão da Veiga and K. Lipnikov, A mimetic discretization of the Stokes problem with selected edge bubbles, SIAM J. Sci. Comput., 32 (2010), 875-893.  doi: 10.1137/090767029.

[20]

L. Beirão da VeigaK. Lipnikov and G. Manzini, Error analysis for a mimetic discretization of the steady Stokes problem on polyhedral meshes, SIAM J. Numer. Anal., 48 (2010), 1419-1443.  doi: 10.1137/090757411.

[21]

L. Beirão da Veiga, K. Lipnikov and G. Manzini, The Mimetic Finite Difference Method, volume 11, MS & A. Modeling, Simulations and Applications, Springer, I edition, 2014.

[22]

L. Beirão da VeigaC. Lovadina and D. Mora, A virtual element method for elastic and inelastic problems on polytope meshes, Comput. Methods Appl. Mech. Engrg., 295 (2015), 327-346.  doi: 10.1016/j.cma.2015.07.013.

[23]

L. Beirão da VeigaC. Lovadina and G. Vacca, Divergence free virtual elements for the Stokes problem on polygonal meshes, ESAIM Math. Model. Numer., 51 (2017), 509-535.  doi: 10.1051/m2an/2016032.

[24]

L. Beirão da VeigaC. Lovadina and G. Vacca, Virtual elements for the Navier-Stokes problem on polygonal meshes, SIAM J. Numer. Anal., 56 (2018), 1210-1242.  doi: 10.1137/17M1132811.

[25]

L. Beirão da Veiga and G. Manzinim, A virtual element method with arbitrary regularity, IMA J. Numer. Anal., 34 (2014), 759-781.  doi: 10.1093/imanum/drt018.

[26]

L. Beirão da Veiga and G. Manzini, Residual a posteriori error estimation for the virtual element method for elliptic problems, ESAIM Math. Model. Numer. Anal., 49 (2015), 577-599.  doi: 10.1051/m2an/2014047.

[27]

L. Beirão da VeigaG. Manzini and L. Mascotto, A posteriori error estimation and adaptivity in hp virtual elements, Numer. Math., 143 (2019), 139-175.  doi: 10.1007/s00211-019-01054-6.

[28]

L. Beirão da VeigaD. Mora and G. Vacca, The Stokes complex for virtual elements with application to Navier–Stokes flows, J. Sci. Comput., 81 (2019), 990-1018.  doi: 10.1007/s10915-019-01049-3.

[29]

M. F. BenedettoS. Berrone and A. Borio, The virtual element method for underground flow simulations in fractured data, Advances in Discretization Methods, SEMA SIMAI Springer Ser., 12 (2016), 167-186. 

[30]

M. F. BenedettoS. BerroneS. Pieraccini and S. Scialò, The virtual element method for discrete fracture network simulations, Comput. Methods Appl. Mech. Engrg., 280 (2014), 135-156.  doi: 10.1016/j.cma.2014.07.016.

[31]

E. BenvenutiA. ChiozziG. Manzini and N. Sukumar, Extended virtual element method for the Laplace problem with singularities and discontinuities, Comput. Methods Appl. Mech. Engrg., 356 (2019), 571-597.  doi: 10.1016/j.cma.2019.07.028.

[32]

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.

[33]

S. BerroneA. Borio and G. Manzini, SUPG stabilization for the nonconforming virtual element method for advection-diffusion–reaction equations, Comput. Methods Appl. Mech. Engrg., 340 (2018), 500-529.  doi: 10.1016/j.cma.2018.05.027.

[34]

S. BerroneA. Borio and S. Scialò, A posteriori error estimate for a PDE-constrained optimization formulation for the flow in DFNs, SIAM J. Numer. Anal., 54 (2016), 242-261.  doi: 10.1137/15M1014760.

[35]

S. BerroneS. Pieraccini and S. Scialò, Towards effective flow simulations in realistic discrete fracture networks, J. Comput. Phys., 310 (2016), 181-201.  doi: 10.1016/j.jcp.2016.01.009.

[36]

S. BerroneS. PieracciniS. Scialò and F. Vicini, A parallel solver for large scale DFN flow simulations, SIAM J. Sci. Comput., 37 (2015), C285-C306.  doi: 10.1137/140984014.

[37]

D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications, Springer Series in Computational Mathematics, 44. Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-36519-5.

[38]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, 15. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4757-4338-8.

[39]

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.

[40]

F. BrezziK. Lipnikov and M. Shashkov, Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes with curved faces, Math. Models Methods Appl. Sci., 16 (2006), 275-297.  doi: 10.1142/S0218202506001157.

[41]

F. BrezziK. LipnikovM. Shashkov and V. Simoncini, A new discretization methodology for diffusion problems on generalized polyhedral meshes, Comput. Methods Appl. Mech. Engrg., 196 (2007), 3682-3692.  doi: 10.1016/j.cma.2006.10.028.

[42]

F. Brezzi and L. D. Marini, Virtual element methods for plate bending problems, Comput. Methods Appl. Mech. Engrg., 253 (2013), 455-462.  doi: 10.1016/j.cma.2012.09.012.

[43]

Z. CaiC. TongP. S. Vassilevski and C. Wang, Mixed finite element methods for incompressible flow: Stationary Stokes equations, Numer. Methods Partial Differ. Equ., 26 (2010), 957-978.  doi: 10.1002/num.20467.

[44]

J. Campbell and M. Shashkov, A tensor artificial viscosity using a mimetic finite difference algorithm, J. Comput. Phys., 172 (2001), 739-765.  doi: 10.1006/jcph.2001.6856.

[45]

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.

[46]

A. CangianiV. Gyrya and G. Manzini, The non-conforming virtual element method for the Stokes equations, SIAM J. Numer. Anal., 54 (2016), 3411-3435.  doi: 10.1137/15M1049531.

[47]

A. Cangiani, V. Gyya, G. Manzini and O. Sutton, Virtual element methods for elliptic problems on polygonal meshes, Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics, CRC Press, Boca Raton, FL, 263 (2018), 263–279.

[48]

A. CangianiG. Manzini and A. Russo, Convergence analysis of a mimetic finite difference method for elliptic problems, SIAM J. Numer. Anal., 47 (2009), 2612-2637.  doi: 10.1137/080717560.

[49]

A. CangianiG. ManziniA. Russo and N. Sukumar, Hourglass stabilization and the virtual element method, Internat. J. Numer. Methods Engrg., 102 (2015), 404-436.  doi: 10.1002/nme.4854.

[50]

A. CangianiG. Manzini and O. Sutton, Conforming and nonconforming virtual element methods for elliptic problems, IMA J. Numer. Anal., 37 (2017), 1317-1354.  doi: 10.1093/imanum/drw036.

[51]

O. CertikF. GardiniG. ManziniL. Mascotto and G. Vacca, The p- and hp-versions of the virtual element method for elliptic eigenvalue problems, Comput. Math. Appl., 79 (2020), 2035-2056.  doi: 10.1016/j.camwa.2019.10.018.

[52]

O. CertikF. GardiniG. Manzini and G. Vacca, The virtual element method for eigenvalue problems with potential terms on polytopic meshes, Appl. Math., 63 (2018), 333-365.  doi: 10.21136/AM.2018.0093-18.

[53]

A. Chernov, C. Marcati and L. Mascotto, p- and hp- virtual elements for the Stokes problem, Adv. Comput. Math., 47 (2021), Paper No. 24, 31 pp. doi: 10.1007/s10444-020-09831-w.

[54]

M. Crouzeix and P. A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations, Recherche Opérationnelle Sér. Rouge, 7 (1973), 33-75. 

[55]

D. A. Di PietroJ. Droniou and G. Manzini, Discontinuous skeletal gradient discretisation methods on polytopal meshes, J. Comput. Phys., 355 (2018), 397-425.  doi: 10.1016/j.jcp.2017.11.018.

[56]

F. GardiniG. Manzini and G. Vacca, The nonconforming virtual element method for eigenvalue problems, ESAIM Math. Model. Numer. Anal., 53 (2019), 749-774.  doi: 10.1051/m2an/2018074.

[57]

V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Mathematics, 749. Springer-Verlag, Berlin-New York, 1979.

[58]

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

[59]

R. M. Höfer, Sedimentation of inertialess particles in Stokes flows, Commun. Math. Phys., 360 (2018), 55-101.  doi: 10.1007/s00220-018-3131-y.

[60]

J. Hyman and M. Shashkov, Mimetic discretizations for Maxwell's equations and the equations of magnetic diffusion, PIER, 32 (2001), 89-121. 

[61]

H. Kitahata, N. Yoshinaga, K. H. Nagai and Y. Sumino, 3 - Dynamics of Droplets, Pattern Formations and Oscillatory Phenomena, (2013), 85–118. doi: 10.1016/B978-0-12-397014-5.00003-1.

[62]

Y. Kuznetsov and S. Repin, New mixed finite element method on polygonal and polyhedral meshes, Russian J. Numer. Anal. Math. Modelling, 18 (2003), 261-278.  doi: 10.1515/156939803322380846.

[63]

S. Linden, L. Cheng and A. Wiegmann, Specialized Methods for Direct Numerical Simulations in Porous Media, Technical Report Report M2M-2018-01, Math2Market GmbH, Kaiserslautern, Germany, 2018.

[64]

K. LipnikovG. Manzini and M. Shashkov, Mimetic finite difference method, J. Comput. Phys., 257 (2014), 1163-1227.  doi: 10.1016/j.jcp.2013.07.031.

[65]

G. ManziniA. Russo and N. Sukumar, New perspectives on polygonal and polyhedral finite element methods, Math. Models Methods Appl. Sci, 24 (2014), 1665-1699.  doi: 10.1142/S0218202514400065.

[66]

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.

[67]

D. MoraG. Rivera and R. Rodríguez, A virtual element method for the Steklov eigenvalue problem, Math. Models Methods Appl. Sci., 25 (2015), 1421-1445.  doi: 10.1142/S0218202515500372.

[68]

S. NatarajanS. P. A. Bordas and E. T. Ooi, Virtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods, Internat. J. Numer. Methods Engrg., 104 (2015), 1173-1199.  doi: 10.1002/nme.4965.

[69]

G. H. Paulino and A. L. Gain, Bridging art and engineering using Escher-based virtual elements, Struct. Multidiscip. Optim., 51 (2015), 867-883.  doi: 10.1007/s00158-014-1179-7.

[70]

I. PerugiaP. Pietra and A. Russo, A plane wave virtual element method for the Helmholtz problem, ESAIM Math. Model. Numer. Anal., 50 (2016), 783-808.  doi: 10.1051/m2an/2015066.

[71]

J. Qin, On the Convergence of some Low Order Mixed Finite Elements for Incompressible Fluids, The Pennsylvania State University, 1994.

[72]

J. P. SmithA. C. BarbatiS. M. SantanaJ. P. Gleghorn and B. J. Kirby, Microfluidic transport in microdevices for rare cell capture, Electrophoresis, 33 (2012), 3133-3142.  doi: 10.1002/elps.201200263.

[73]

N. Sukumar and A. Tabarraei, Conforming polygonal finite elements, Internat. J. Numer. Methods Engrg., 61 (2004), 2045-2066.  doi: 10.1002/nme.1141.

[74]

A. Tabarraei and N. Sukumar, Extended finite element method on polygonal and quadtree meshes, Comput. Methods Appl. Mech. Engrg., 197 (2007), 425-438.  doi: 10.1016/j.cma.2007.08.013.

[75]

C. TalischiG. H. PaulinoA. Pereira and I. F. M. Menezes, Polygonal finite elements for topology optimization: A unifying paradigm, Int. J. Numer. Methods Eng., 82 (2010), 671-698.  doi: 10.1002/nme.2763.

[76]

G. Vacca and L. Beirão da Veiga, Virtual element methods for parabolic problems on polygonal meshes, Numer. Methods Partial Differential Equations, 31 (2015), 2110-2134.  doi: 10.1002/num.21982.

[77]

E. Wachspress, Rational Bases and Generalized Barycentrics: Applications to Finite Elements and Graphics, Springer, Cham, 2016. doi: 10.1007/978-3-319-21614-0.

[78]

P. WriggersW. T. Rust and B. D. Reddy, A virtual element method for contact, Comput. Mech., 58 (2016), 1039-1050.  doi: 10.1007/s00466-016-1331-x.

[79]

J. ZhaoS. Chen and B. Zhang, The nonconforming virtual element method for plate bending problems, Math. Models Methods Appl. Sci., 26 (2016), 1671-1687.  doi: 10.1142/S021820251650041X.

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003.

[2]

B. AhmadA. AlsaediF. BrezziL. D. Marini and A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl., 66 (2013), 376-391.  doi: 10.1016/j.camwa.2013.05.015.

[3]

P. F. AntoniettiL. Beirão da VeigaD. Mora and M. Verani, A stream virtual element formulation of the Stokes problem on polygonal meshes, SIAM J. Numer. Anal., 52 (2014), 386-404.  doi: 10.1137/13091141X.

[4]

P. F. AntoniettiG. Manzini and M. Verani, The fully nonconforming Virtual Element method for biharmonic problems, Math. Models Methods Appl. Sci., 28 (2018), 387-407.  doi: 10.1142/S0218202518500100.

[5]

P. F. AntoniettiG. Manzini and M. Verani, The conforming virtual element method for polyharmonic problems, Comput. Math. Appl., 79 (2020), 2021-2034.  doi: 10.1016/j.camwa.2019.09.022.

[6]

B. Ayuso de DiosK. Lipnikov and G. Manzini, The non-conforming virtual element method, ESAIM Math. Model. Numer. Anal., 50 (2016), 879-904.  doi: 10.1051/m2an/2015090.

[7]

B. Bang and D. Lukkassen, Application of homogenization theory related to Stokes flow in porous media, Appl. Math., 44 (1999), 309-319.  doi: 10.1023/A:1023084614058.

[8]

L. Beirão da VeigaF. BrezziA. CangianiG. ManziniL. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23 (2013), 199-214.  doi: 10.1142/S0218202512500492.

[9]

L. Beirão da VeigaF. Brezzi and L. D. Marini, Virtual elements for linear elasticity problems, SIAM J. Numer. Anal., 51 (2013), 794-812.  doi: 10.1137/120874746.

[10]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, The hitchhiker's guide to the virtual element method, Math. Models Methods Appl. Sci, 24 (2014), 1541-1573.  doi: 10.1142/S021820251440003X.

[11]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, Virtual element methods for general second order elliptic problems on polygonal meshes, Math. Models Methods Appl. Sci., 26 (2016), 729-750.  doi: 10.1142/S0218202516500160.

[12]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, H(div) and H(curl)-conforming VEM, Numer. Math., 133 (2016), 303-332.  doi: 10.1007/s00211-015-0746-1.

[13]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, Mixed virtual element methods for general second order elliptic problems on polygonal meshes, ESAIM Math. Model. Numer. Anal., 50 (2016), 727-747.  doi: 10.1051/m2an/2015067.

[14]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, Serendipity nodal VEM spaces, Comput. Fluids, 141 (2016), 2-12.  doi: 10.1016/j.compfluid.2016.02.015.

[15]

L. Beirão da VeigaF. BrezziL. D. Marini and A. Russo, Virtual element methods for general second order elliptic problems on polygonal meshes, Math. Models Methods Appl. Sci., 26 (2016), 729-750.  doi: 10.1142/S0218202516500160.

[16]

L. Beirão da VeigaA. ChernovL. Mascotto and A. Russo, Basic principles of hp virtual elements on quasiuniform meshes, Math. Models Methods Appl. Sci., 26 (2016), 1567-1598.  doi: 10.1142/S021820251650038X.

[17]

L. Beirão da VeigaF. Dassi and G. Vacca, The Stokes complex for virtual elements in three dimensions, Math. Models Methods Appl. Sci., 30 (2020), 477-512.  doi: 10.1142/S0218202520500128.

[18]

L. Beirão da VeigaV. GyryaK. Lipnikov and G. Manzini, Mimetic finite difference method for the Stokes problem on polygonal meshes, J. Comput. Phys., 228 (2009), 7215-7232.  doi: 10.1016/j.jcp.2009.06.034.

[19]

L. Beirão da Veiga and K. Lipnikov, A mimetic discretization of the Stokes problem with selected edge bubbles, SIAM J. Sci. Comput., 32 (2010), 875-893.  doi: 10.1137/090767029.

[20]

L. Beirão da VeigaK. Lipnikov and G. Manzini, Error analysis for a mimetic discretization of the steady Stokes problem on polyhedral meshes, SIAM J. Numer. Anal., 48 (2010), 1419-1443.  doi: 10.1137/090757411.

[21]

L. Beirão da Veiga, K. Lipnikov and G. Manzini, The Mimetic Finite Difference Method, volume 11, MS & A. Modeling, Simulations and Applications, Springer, I edition, 2014.

[22]

L. Beirão da VeigaC. Lovadina and D. Mora, A virtual element method for elastic and inelastic problems on polytope meshes, Comput. Methods Appl. Mech. Engrg., 295 (2015), 327-346.  doi: 10.1016/j.cma.2015.07.013.

[23]

L. Beirão da VeigaC. Lovadina and G. Vacca, Divergence free virtual elements for the Stokes problem on polygonal meshes, ESAIM Math. Model. Numer., 51 (2017), 509-535.  doi: 10.1051/m2an/2016032.

[24]

L. Beirão da VeigaC. Lovadina and G. Vacca, Virtual elements for the Navier-Stokes problem on polygonal meshes, SIAM J. Numer. Anal., 56 (2018), 1210-1242.  doi: 10.1137/17M1132811.

[25]

L. Beirão da Veiga and G. Manzinim, A virtual element method with arbitrary regularity, IMA J. Numer. Anal., 34 (2014), 759-781.  doi: 10.1093/imanum/drt018.

[26]

L. Beirão da Veiga and G. Manzini, Residual a posteriori error estimation for the virtual element method for elliptic problems, ESAIM Math. Model. Numer. Anal., 49 (2015), 577-599.  doi: 10.1051/m2an/2014047.

[27]

L. Beirão da VeigaG. Manzini and L. Mascotto, A posteriori error estimation and adaptivity in hp virtual elements, Numer. Math., 143 (2019), 139-175.  doi: 10.1007/s00211-019-01054-6.

[28]

L. Beirão da VeigaD. Mora and G. Vacca, The Stokes complex for virtual elements with application to Navier–Stokes flows, J. Sci. Comput., 81 (2019), 990-1018.  doi: 10.1007/s10915-019-01049-3.

[29]

M. F. BenedettoS. Berrone and A. Borio, The virtual element method for underground flow simulations in fractured data, Advances in Discretization Methods, SEMA SIMAI Springer Ser., 12 (2016), 167-186. 

[30]

M. F. BenedettoS. BerroneS. Pieraccini and S. Scialò, The virtual element method for discrete fracture network simulations, Comput. Methods Appl. Mech. Engrg., 280 (2014), 135-156.  doi: 10.1016/j.cma.2014.07.016.

[31]

E. BenvenutiA. ChiozziG. Manzini and N. Sukumar, Extended virtual element method for the Laplace problem with singularities and discontinuities, Comput. Methods Appl. Mech. Engrg., 356 (2019), 571-597.  doi: 10.1016/j.cma.2019.07.028.

[32]

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.

[33]

S. BerroneA. Borio and G. Manzini, SUPG stabilization for the nonconforming virtual element method for advection-diffusion–reaction equations, Comput. Methods Appl. Mech. Engrg., 340 (2018), 500-529.  doi: 10.1016/j.cma.2018.05.027.

[34]

S. BerroneA. Borio and S. Scialò, A posteriori error estimate for a PDE-constrained optimization formulation for the flow in DFNs, SIAM J. Numer. Anal., 54 (2016), 242-261.  doi: 10.1137/15M1014760.

[35]

S. BerroneS. Pieraccini and S. Scialò, Towards effective flow simulations in realistic discrete fracture networks, J. Comput. Phys., 310 (2016), 181-201.  doi: 10.1016/j.jcp.2016.01.009.

[36]

S. BerroneS. PieracciniS. Scialò and F. Vicini, A parallel solver for large scale DFN flow simulations, SIAM J. Sci. Comput., 37 (2015), C285-C306.  doi: 10.1137/140984014.

[37]

D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications, Springer Series in Computational Mathematics, 44. Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-36519-5.

[38]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, 15. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4757-4338-8.

[39]

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.

[40]

F. BrezziK. Lipnikov and M. Shashkov, Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes with curved faces, Math. Models Methods Appl. Sci., 16 (2006), 275-297.  doi: 10.1142/S0218202506001157.

[41]

F. BrezziK. LipnikovM. Shashkov and V. Simoncini, A new discretization methodology for diffusion problems on generalized polyhedral meshes, Comput. Methods Appl. Mech. Engrg., 196 (2007), 3682-3692.  doi: 10.1016/j.cma.2006.10.028.

[42]

F. Brezzi and L. D. Marini, Virtual element methods for plate bending problems, Comput. Methods Appl. Mech. Engrg., 253 (2013), 455-462.  doi: 10.1016/j.cma.2012.09.012.

[43]

Z. CaiC. TongP. S. Vassilevski and C. Wang, Mixed finite element methods for incompressible flow: Stationary Stokes equations, Numer. Methods Partial Differ. Equ., 26 (2010), 957-978.  doi: 10.1002/num.20467.

[44]

J. Campbell and M. Shashkov, A tensor artificial viscosity using a mimetic finite difference algorithm, J. Comput. Phys., 172 (2001), 739-765.  doi: 10.1006/jcph.2001.6856.

[45]

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.

[46]

A. CangianiV. Gyrya and G. Manzini, The non-conforming virtual element method for the Stokes equations, SIAM J. Numer. Anal., 54 (2016), 3411-3435.  doi: 10.1137/15M1049531.

[47]

A. Cangiani, V. Gyya, G. Manzini and O. Sutton, Virtual element methods for elliptic problems on polygonal meshes, Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics, CRC Press, Boca Raton, FL, 263 (2018), 263–279.

[48]

A. CangianiG. Manzini and A. Russo, Convergence analysis of a mimetic finite difference method for elliptic problems, SIAM J. Numer. Anal., 47 (2009), 2612-2637.  doi: 10.1137/080717560.

[49]

A. CangianiG. ManziniA. Russo and N. Sukumar, Hourglass stabilization and the virtual element method, Internat. J. Numer. Methods Engrg., 102 (2015), 404-436.  doi: 10.1002/nme.4854.

[50]

A. CangianiG. Manzini and O. Sutton, Conforming and nonconforming virtual element methods for elliptic problems, IMA J. Numer. Anal., 37 (2017), 1317-1354.  doi: 10.1093/imanum/drw036.

[51]

O. CertikF. GardiniG. ManziniL. Mascotto and G. Vacca, The p- and hp-versions of the virtual element method for elliptic eigenvalue problems, Comput. Math. Appl., 79 (2020), 2035-2056.  doi: 10.1016/j.camwa.2019.10.018.

[52]

O. CertikF. GardiniG. Manzini and G. Vacca, The virtual element method for eigenvalue problems with potential terms on polytopic meshes, Appl. Math., 63 (2018), 333-365.  doi: 10.21136/AM.2018.0093-18.

[53]

A. Chernov, C. Marcati and L. Mascotto, p- and hp- virtual elements for the Stokes problem, Adv. Comput. Math., 47 (2021), Paper No. 24, 31 pp. doi: 10.1007/s10444-020-09831-w.

[54]

M. Crouzeix and P. A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations, Recherche Opérationnelle Sér. Rouge, 7 (1973), 33-75. 

[55]

D. A. Di PietroJ. Droniou and G. Manzini, Discontinuous skeletal gradient discretisation methods on polytopal meshes, J. Comput. Phys., 355 (2018), 397-425.  doi: 10.1016/j.jcp.2017.11.018.

[56]

F. GardiniG. Manzini and G. Vacca, The nonconforming virtual element method for eigenvalue problems, ESAIM Math. Model. Numer. Anal., 53 (2019), 749-774.  doi: 10.1051/m2an/2018074.

[57]

V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Mathematics, 749. Springer-Verlag, Berlin-New York, 1979.

[58]

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

[59]

R. M. Höfer, Sedimentation of inertialess particles in Stokes flows, Commun. Math. Phys., 360 (2018), 55-101.  doi: 10.1007/s00220-018-3131-y.

[60]

J. Hyman and M. Shashkov, Mimetic discretizations for Maxwell's equations and the equations of magnetic diffusion, PIER, 32 (2001), 89-121. 

[61]

H. Kitahata, N. Yoshinaga, K. H. Nagai and Y. Sumino, 3 - Dynamics of Droplets, Pattern Formations and Oscillatory Phenomena, (2013), 85–118. doi: 10.1016/B978-0-12-397014-5.00003-1.

[62]

Y. Kuznetsov and S. Repin, New mixed finite element method on polygonal and polyhedral meshes, Russian J. Numer. Anal. Math. Modelling, 18 (2003), 261-278.  doi: 10.1515/156939803322380846.

[63]

S. Linden, L. Cheng and A. Wiegmann, Specialized Methods for Direct Numerical Simulations in Porous Media, Technical Report Report M2M-2018-01, Math2Market GmbH, Kaiserslautern, Germany, 2018.

[64]

K. LipnikovG. Manzini and M. Shashkov, Mimetic finite difference method, J. Comput. Phys., 257 (2014), 1163-1227.  doi: 10.1016/j.jcp.2013.07.031.

[65]

G. ManziniA. Russo and N. Sukumar, New perspectives on polygonal and polyhedral finite element methods, Math. Models Methods Appl. Sci, 24 (2014), 1665-1699.  doi: 10.1142/S0218202514400065.

[66]

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.

[67]

D. MoraG. Rivera and R. Rodríguez, A virtual element method for the Steklov eigenvalue problem, Math. Models Methods Appl. Sci., 25 (2015), 1421-1445.  doi: 10.1142/S0218202515500372.

[68]

S. NatarajanS. P. A. Bordas and E. T. Ooi, Virtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods, Internat. J. Numer. Methods Engrg., 104 (2015), 1173-1199.  doi: 10.1002/nme.4965.

[69]

G. H. Paulino and A. L. Gain, Bridging art and engineering using Escher-based virtual elements, Struct. Multidiscip. Optim., 51 (2015), 867-883.  doi: 10.1007/s00158-014-1179-7.

[70]

I. PerugiaP. Pietra and A. Russo, A plane wave virtual element method for the Helmholtz problem, ESAIM Math. Model. Numer. Anal., 50 (2016), 783-808.  doi: 10.1051/m2an/2015066.

[71]

J. Qin, On the Convergence of some Low Order Mixed Finite Elements for Incompressible Fluids, The Pennsylvania State University, 1994.

[72]

J. P. SmithA. C. BarbatiS. M. SantanaJ. P. Gleghorn and B. J. Kirby, Microfluidic transport in microdevices for rare cell capture, Electrophoresis, 33 (2012), 3133-3142.  doi: 10.1002/elps.201200263.

[73]

N. Sukumar and A. Tabarraei, Conforming polygonal finite elements, Internat. J. Numer. Methods Engrg., 61 (2004), 2045-2066.  doi: 10.1002/nme.1141.

[74]

A. Tabarraei and N. Sukumar, Extended finite element method on polygonal and quadtree meshes, Comput. Methods Appl. Mech. Engrg., 197 (2007), 425-438.  doi: 10.1016/j.cma.2007.08.013.

[75]

C. TalischiG. H. PaulinoA. Pereira and I. F. M. Menezes, Polygonal finite elements for topology optimization: A unifying paradigm, Int. J. Numer. Methods Eng., 82 (2010), 671-698.  doi: 10.1002/nme.2763.

[76]

G. Vacca and L. Beirão da Veiga, Virtual element methods for parabolic problems on polygonal meshes, Numer. Methods Partial Differential Equations, 31 (2015), 2110-2134.  doi: 10.1002/num.21982.

[77]

E. Wachspress, Rational Bases and Generalized Barycentrics: Applications to Finite Elements and Graphics, Springer, Cham, 2016. doi: 10.1007/978-3-319-21614-0.

[78]

P. WriggersW. T. Rust and B. D. Reddy, A virtual element method for contact, Comput. Mech., 58 (2016), 1039-1050.  doi: 10.1007/s00466-016-1331-x.

[79]

J. ZhaoS. Chen and B. Zhang, The nonconforming virtual element method for plate bending problems, Math. Models Methods Appl. Sci., 26 (2016), 1671-1687.  doi: 10.1142/S021820251650041X.

Figure 1.  Degrees of freedom of each component of the virtual element vector-valued fields of $ {\bf{V}}^{h}_{k}( {\rm{E}}) $ (left) and the scalar polynomial fields of $ Q^{ h}_{\underline{k}}( {\rm{E}}) $ (right) of an hexagonal element for the accuracy degrees $ k = 1,2,3 $ and $ \underline{k} = k-1 $. Vertex values and edge polynomial moments are marked by a circular bullet; cell polynomial moments are marked by a square bullet
Figure 2.  Base meshes (top row) and first refinement meshes (bottom row) of the three mesh families used in the general convergence tests: $ \mathcal{M}{1} $: randomly quadrilateral meshes; $ \mathcal{M}{2} $: general polygonal meshes; $ \mathcal{M}{3} $: concave element meshes
Figure 3.  Base meshes (top row) and first refinement meshes (bottom row) of the three mesh families used to investigate convergence and stability of the lowest-order scheme: $ \mathcal{M}{4} $: diagonal triangle meshes; $ \mathcal{M}{5} $: criss-cross triangle meshes; $ \mathcal{M}{6} $: square meshes
Figure 4.  Convergence curves for $ k = 1,\ldots,6 $ and $ \underline{k} = k-1 $ versus the mesh size parameter $ h $ for the velocity approximation measured using the energy norm (55) (top panels) and the $ L^2 $-norm (56) (mid panels), and for the pressure approximation measured using the $ L^2 $-norm (57) (bottom panels). Blue lines with circles represent the error curves using the enhanced virtual element space (14), and, accordingly, the right-hand is approximated by using the projection operator $ \Pi^{0, {\rm{E}}}_{{k}} $. The mesh families used in each calculations are shown in the left corner of each panel. The expected convergence slopes and rates are shown by the triangles and corresponding numerical labels
Figure 5.  Convergence curves for $ k = 1,\ldots,6 $ and $ \underline{k} = k-1 $ versus the mesh size parameter $ h $ for the velocity approximation measured using the energy norm (55) (top panels) and the $ L^2 $-norm (56) (mid panels), and for the pressure approximation measured using the $ L^2 $-norm (57) (bottom panels). Blue lines with circles represent the error curves using the virtual element space (13), and, accordingly, the right-hand is approximated by using the projection operator $ \Pi^{0, {\rm{E}}}_{{ \bar{k}}} $ with $ \bar{k} = max(0,k-2) $. A loss of accuracy for $ k = 2 $ in the $ L^2 $-norm error curves is visible. The mesh families used in each calculations are shown in the left corner of each panel. The expected convergence slopes and rates are shown by the triangles and corresponding numeric labels
Figure 6.  Values of the inf-sup constant $ \beta $ versus the mesh size parameter $ h $. The lines with circles represent the values of $ \beta $ with $ k = 1 $ and $ \underline{k} = 0 $. The mesh families used in each calculations are shown in the bottom-left corner of each panel
Figure 7.  Values of the inf-sup constant $ \beta $ versus the mesh size parameter $ h $. Blue lines with circles represent the values of $ \beta $ with $ k = 2 $ and $ \underline{k} = 0 $. Red lines with circles represent the values of $ \beta $ when $ k = 2 $ and $ \underline{k} = 1 $. Black, green and magenta lines with triangles are associate to $ k = 3 $ and $ \underline{k} = 0 $, $ \underline{k} = 1 $ and $ \underline{k} = 2 $, respectively. The mesh families used in each calculations are shown in the bottom-left corner of each panel
Figure 8.  Convergence curves for $ k = 2,\ldots,6 $, and $ \underline{k} = k-1 $ versus the mesh size parameter $ h $ for the velocity approximation measured using the energy norm (55) (top panels) and the $ L^2 $-norm (56) (mid panels), and for the pressure approximation measured using the $ L^2 $-norm (57) (bottom panels). The results with $ k = 1 $ are not reported because there is no convergence. The lines with circles represent the error curves using the enhanced virtual element space (14). The right-hand side is approximated by using the projection operator $ \Pi^{0, {\rm{E}}}_{{k}} $, i.e., the "enhanced" definition of the virtual element space given in (14). The mesh families used in each calculations are shown in the left corner of each panel. The expected convergence slopes and rates are shown by the triangles and corresponding numerical labels
Figure 9.  Convergence curves for $ k = 2 $, $ \underline{k} = 0 $, versus the mesh size parameter $ h $ for the velocity approximation measured using the energy norm (55) (top panels) and the $ L^2 $-norm (56) (mid panels), and for the pressure approximation measured using the $ L^2 $-norm (57) (bottom panels). Blue lines with circles represent the error curves for the formulation using the enhanced virtual element space (14) with the right-hand side approximated by using the projection operator $ \Pi^{0}_{{2}} $. Red lines with circles represent the error curve with the right-hand side approximated by using the projection operator $ \Pi^{0}_{{0}} $. The mesh families used in each calculations are shown in the left corner of each panel. The convergence slopes and rates are shown by the triangles and corresponding numeric labels
Table 1.  Diameter $ h $ of each grid of the six mesh families $ \mathcal{M}{1} $-$ \mathcal{M}{6} $
Level $ \mathcal{M}{1} $ $ \mathcal{M}{2} $ $ \mathcal{M}{3} $ $ \mathcal{M}{4} $ $ \mathcal{M}{5} $ $ \mathcal{M}{6} $
1 $ 3.72 \cdot 10^{-1} $ $ 4.26 \cdot 10^{-1} $ $ 3.81 \cdot 10^{-1} $ $ 7.07 \cdot 10^{-1} $ $ 5.00 \cdot 10^{-1} $ $ 3.53 \cdot 10^{-1} $
2 $ 1.99 \cdot 10^{-1} $ $ 2.50 \cdot 10^{-1} $ $ 1.91 \cdot 10^{-1} $ $ 3.53 \cdot 10^{-1} $ $ 2.50 \cdot 10^{-1} $ $ 1.77 \cdot 10^{-1} $
3 $ 1.01 \cdot 10^{-1} $ $ 1.25 \cdot 10^{-1} $ $ 9.54 \cdot 10^{-2} $ $ 1.77 \cdot 10^{-1} $ $ 1.25 \cdot 10^{-1} $ $ 8.84 \cdot 10^{-2} $
4 $ 5.17 \cdot 10^{-2} $ $ 6.21 \cdot 10^{-2} $ $ 4.77 \cdot 10^{-2} $ $ 8.84 \cdot 10^{-2} $ $ 6.25 \cdot 10^{-1} $ $ 4.42 \cdot 10^{-2} $
5 $ 2.61 \cdot 10^{-2} $ $ 3.41 \cdot 10^{-2} $ $ 2.38 \cdot 10^{-2} $ $ 4.42 \cdot 10^{-2} $ $ 3.12 \cdot 10^{-2} $ $ 2.21 \cdot 10^{-2} $
Level $ \mathcal{M}{1} $ $ \mathcal{M}{2} $ $ \mathcal{M}{3} $ $ \mathcal{M}{4} $ $ \mathcal{M}{5} $ $ \mathcal{M}{6} $
1 $ 3.72 \cdot 10^{-1} $ $ 4.26 \cdot 10^{-1} $ $ 3.81 \cdot 10^{-1} $ $ 7.07 \cdot 10^{-1} $ $ 5.00 \cdot 10^{-1} $ $ 3.53 \cdot 10^{-1} $
2 $ 1.99 \cdot 10^{-1} $ $ 2.50 \cdot 10^{-1} $ $ 1.91 \cdot 10^{-1} $ $ 3.53 \cdot 10^{-1} $ $ 2.50 \cdot 10^{-1} $ $ 1.77 \cdot 10^{-1} $
3 $ 1.01 \cdot 10^{-1} $ $ 1.25 \cdot 10^{-1} $ $ 9.54 \cdot 10^{-2} $ $ 1.77 \cdot 10^{-1} $ $ 1.25 \cdot 10^{-1} $ $ 8.84 \cdot 10^{-2} $
4 $ 5.17 \cdot 10^{-2} $ $ 6.21 \cdot 10^{-2} $ $ 4.77 \cdot 10^{-2} $ $ 8.84 \cdot 10^{-2} $ $ 6.25 \cdot 10^{-1} $ $ 4.42 \cdot 10^{-2} $
5 $ 2.61 \cdot 10^{-2} $ $ 3.41 \cdot 10^{-2} $ $ 2.38 \cdot 10^{-2} $ $ 4.42 \cdot 10^{-2} $ $ 3.12 \cdot 10^{-2} $ $ 2.21 \cdot 10^{-2} $
Table 2.  Number of elements $ N_{el} $ and vertices $ N $ of each grid of the three mesh families $ \mathcal{M}{1} $$ \mathcal{M}{3} $
Level $ \mathcal{M}{1} $ $ \mathcal{M}{2} $ $ \mathcal{M}{3} $
$ N_{el} $ $ N $ $ N_{el} $ $ N $ $ N_{el} $ $ N $
1 16 25 22 46 16 73
2 64 81 84 171 64 305
3 256 289 312 628 256 1249
4 1024 1089 1202 2406 1024 5057
5 4096 4225 4772 9547 4096 20353
Level $ \mathcal{M}{1} $ $ \mathcal{M}{2} $ $ \mathcal{M}{3} $
$ N_{el} $ $ N $ $ N_{el} $ $ N $ $ N_{el} $ $ N $
1 16 25 22 46 16 73
2 64 81 84 171 64 305
3 256 289 312 628 256 1249
4 1024 1089 1202 2406 1024 5057
5 4096 4225 4772 9547 4096 20353
Table 3.  Number of elements $ N_{el} $ and vertices $ N $ of each grid of the three mesh families $ \mathcal{M}{4} $-$ \mathcal{M}{6} $
Level $ \mathcal{M}{4} $ $ \mathcal{M}{5} $ $ \mathcal{M}{6} $
$ N_{el} $ $ N $ $ N_{el} $ $ N $ $ N_{el} $ $ N $
1 8 9 16 13 16 25
2 32 25 64 41 64 81
3 128 81 256 145 256 289
4 512 289 1024 545 1024 1089
5 2048 1089 4096 2113 4096 4225
Level $ \mathcal{M}{4} $ $ \mathcal{M}{5} $ $ \mathcal{M}{6} $
$ N_{el} $ $ N $ $ N_{el} $ $ N $ $ N_{el} $ $ N $
1 8 9 16 13 16 25
2 32 25 64 41 64 81
3 128 81 256 145 256 289
4 512 289 1024 545 1024 1089
5 2048 1089 4096 2113 4096 4225
Table 4.  Size, rank and kernel's dimension of matrix B when $ k = 1 $ for the mesh family $ \mathcal{M}{4} $
$ \mathcal{M}{4} $
Level size($ B $) $ rank(B) $ $ kernel(B) $
1 $ 2\times 8 $ 2 6
2 $ 18\times 32 $ 18 14
3 $ 98\times 128 $ 98 30
4 $ 450\times 512 $ 450 62
5 $ 1922\times 2048 $ 1922 126
6 $ 7938\times 8192 $ 7938 254
$ \mathcal{M}{4} $
Level size($ B $) $ rank(B) $ $ kernel(B) $
1 $ 2\times 8 $ 2 6
2 $ 18\times 32 $ 18 14
3 $ 98\times 128 $ 98 30
4 $ 450\times 512 $ 450 62
5 $ 1922\times 2048 $ 1922 126
6 $ 7938\times 8192 $ 7938 254
Table 5.  Size, rank and kernel's dimension of matrix B when $ k = 1 $ for the mesh family $ \mathcal{M}{5} $
$ \mathcal{M}{5} $
Level size($ B $) $ rank(B) $ $ kernel(B) $
1 $ 10\times 16 $ 10 6
2 $ 50\times 64 $ 46 18
3 $ 226\times 256 $ 190 66
4 $ 962\times 1024 $ 766 258
5 $ 3970\times 4096 $ 3070 1026
6 $ 16130\times 16384 $ 12286 4098
$ \mathcal{M}{5} $
Level size($ B $) $ rank(B) $ $ kernel(B) $
1 $ 10\times 16 $ 10 6
2 $ 50\times 64 $ 46 18
3 $ 226\times 256 $ 190 66
4 $ 962\times 1024 $ 766 258
5 $ 3970\times 4096 $ 3070 1026
6 $ 16130\times 16384 $ 12286 4098
Table 6.  Size, rank and kernel's dimension of matrix B when $ k = 1 $ for the mesh family $ \mathcal{M}{6} $
$ \mathcal{M}{6} $
Level size($ B $) $ rank(B) $ $ kernel(B) $
1 $ 18\times 16 $ 14 2
2 $ 98\times 64 $ 62 2
3 $ 450\times 256 $ 254 2
4 $ 1922\times 1024 $ 1022 2
5 $ 7938\times 4096 $ 4094 2
6 $ 32258\times 16384 $ 16382 2
$ \mathcal{M}{6} $
Level size($ B $) $ rank(B) $ $ kernel(B) $
1 $ 18\times 16 $ 14 2
2 $ 98\times 64 $ 62 2
3 $ 450\times 256 $ 254 2
4 $ 1922\times 1024 $ 1022 2
5 $ 7938\times 4096 $ 4094 2
6 $ 32258\times 16384 $ 16382 2
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