×

A divergence free weak virtual element method for the Stokes problem on polytopal meshes. (English) Zbl 1433.76074

Summary: Some virtual element methods on polytopal meshes for the Stokes problem are proposed and analyzed. The pressure is approximated by discontinuous polynomials, while the velocity is discretized by H(div) virtual elements enriched with some tangential polynomials on the element boundaries. A weak symmetric gradient of the velocity is computed using the corresponding degree of freedoms. The main feature of the method is that it exactly preserves the divergence free constraint, and therefore the error estimates for the velocity does not explicitly depend on the pressure.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows

Software:

iFEM; PolyMesher
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Antonietti, P.F., Beirão da Veiga, L., Mora, D., Verani, M.: A stream virtual element formulation of the Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 52(1), 386-404 (2014) · Zbl 1427.76198
[2] Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(1), 199-214 (2013) · Zbl 1416.65433
[3] Beirão da Veiga, L., Brezzi, F., Marini, L.D.: Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51(2), 794-812 (2013) · Zbl 1268.74010
[4] Beirão da Veiga, L., Brezzi, F., Marini, L.D., Russo, A.: H(div) and H(curl)-conforming virtual element methods. Numer. Math. 133(2), 303-332 (2016) · Zbl 1343.65133
[5] Beirão da Veiga, L., Brezzi, F., Marini, L.D., Russo, A.: Serendipity face and edge vem spaces. Rend. Lincei Mat. Appl. 28(1), 143-181 (2017) · Zbl 1395.65139
[6] Beirão da Veiga, L., Lovadina, C., Vacca, G.: Divergence free virtual elements for the Stokes problem on polygonal meshes. ESAIM Math. Model. Numer. Anal. 51(2), 509-535 (2017) · Zbl 1398.76094
[7] Beirão da Veiga, L., Lovadina, C., Vacca, G.: Virtual Elements for the Navier-Stokes Problem on Polygonal Meshes. SIAM J. Numer. Anal. 56(3), 1210-1242 (2018) · Zbl 1397.65302
[8] Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Berlin (2013) · Zbl 1277.65092
[9] Brenner, S.C.: Poincaré-Friedrichs inequalities for piecewise \[H^11\] functions. SIAM J. Numer. Anal. 41(1), 306-324 (2003) · Zbl 1045.65100
[10] Brenner, S.C.: Korn’s inequalities for piecewise \[H^11\] vector fields. Math. Comput. 73(247), 1067-1087 (2004) · Zbl 1055.65118
[11] Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 15th edn. Springer, Berlin (2007)
[12] Brenner, S.C., Sung, L.Y.: Virtual element methods on meshes with small edges or faces. Math. Models Methods Appl. Sci. 28(7), 1291-1336 (2018) · Zbl 1393.65049
[13] Brezzi, F., Falk, R.S., Marini., L.D.: Basic principles of mixed virtual element methods. ESAIM Math. Model. Numer. Anal. 48, 1227-1240 (2014) · Zbl 1299.76130
[14] Cangiani, A., Gyrya, V., Manzini, G.: The nonconforming virtual element method for the Stokes equations. SIAM J. Numer. Anal. 54(6), 3411-3435 (2016) · Zbl 1426.76230
[15] Chen, L.: \[i\] iFEM: An Innovative Finite Element Methods Package in MATLAB. Preprint, University of Maryland, College Park (2008)
[16] Chen, L., Huang, J.: Some error analysis on virtual element methods. Calcolo 55(1), 5 (2018) · Zbl 1448.65223
[17] Chen, L., Wang, M., Zhong, L.: Convergence analysis of triangular MAC schemes for two dimensional Stokes equations. J. Sci. Comput. 63(3), 716-744 (2015) · Zbl 1320.76065
[18] Chen, W., Wang, Y.: Minimal degree H(curl) and H(div) conforming finite elements on polytopal meshes. Math. Comput. 86(307), 2053-2087 (2017) · Zbl 1364.65244
[19] Cockburn, B., Kanschat, G., Schötzau, D.: A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations. J. Sci. Comput. 31(1-2), 61-73 (2007) · Zbl 1151.76527
[20] Cockburn, B., Nguyen, N.C., Peraire., J.: A comparison of HDG methods for Stokes flow. J. Sci. Comput. 45(1-3), 215-237 (2010) · Zbl 1203.76079
[21] Cockburn, B., Sayas, F.-J.: Divergence-conforming HDG methods for Stokes flows. Math. Comput. 83(288), 1571-1598 (2014) · Zbl 1427.76125
[22] Cockburn, B., Shi, K.: Devising HDG methods for Stokes flow: an overview. Comput. Fluids 98, 221-229 (2014) · Zbl 1391.76315
[23] De Dios, B.A., Brezzi, F., Marini, L.D., Xu, J., Zikatanov., L.: A simple preconditioner for a discontinuous Galerkin method for the Stokes problem. J. Sci. Comput. 58(3), 517-547 (2014) · Zbl 1299.76128
[24] Di Pietro, D.A., Ern, A.: A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Methods Appl. Mech. Eng. 283, 1-21 (2015) · Zbl 1423.74876
[25] Di Pietro, D.A., Ern, A., Linke, A., Schieweck., F.: A discontinuous skeletal method for the viscosity-dependent Stokes problem. Comput. Methods Appl. Mech. Eng. 306, 175-195 (2016) · Zbl 1436.76022
[26] Di Pietro, D.A., Lemaire., S.: An extension of the Crouzeix-Raviart space to general meshes with application to quasi-incompressible linear elasticity and Stokes flow. Math. Comput. 84(291), 1-31 (2015) · Zbl 1308.74145
[27] Falk, R.S., Neilan, M.: Stokes complexes and the construction of stable finite elements with pointwise mass conservation. SIAM J. Numer. Anal. 51(2), 1308-1326 (2013) · Zbl 1268.76032
[28] Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations. Springer, Berlin (1986) · Zbl 0585.65077
[29] Guzmán, J., Neilan, M.: Conforming and divergence-free Stokes elements in three dimensions. IMA J. Numer. Anal. 34(4), 1489-1508 (2014) · Zbl 1305.76056
[30] Guzmán, J., Neilan, M.: Conforming and divergence-free Stokes elements on general triangular meshes. Math. Comput. 83(285), 15-36 (2014) · Zbl 1322.76041
[31] John, V., Linke, A., Merdon, C., Neilan, M., Rebholz, L.G.: On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59(3), 492-544 (2017) · Zbl 1426.76275
[32] Layton, W.: Introduction to the Numerical Analysis of Incompressible Viscous Flows, vol. 6. SIAM, Tulsa (2008) · Zbl 1153.76002
[33] Lederer, P.L., Linke, A., Merdon, C., Schöberl., J.: Divergence-free reconstruction operators for pressure-robust Stokes discretizations with continuous pressure finite elements. SIAM Rev. 59(3), 492-544 (2017) · Zbl 1457.65202
[34] Lipnikov, K., Manzini, G., Shashkov, M.: Mimetic finite difference method. J. Comput. Phys. 257, 1163-1227 (2014) · Zbl 1352.65420
[35] Liu, X., Li, J., Chen, Z.: A nonconforming virtual element method for the Stokes problem on general meshes. Comput. Methods Appl. Mech. Eng. 320, 694-711 (2017) · Zbl 1439.76085
[36] Mardal, K.A., Tai, X.C., Winther., R.: A robust finite element method for Darcy-Stokes flow. SIAM J. Numer. Anal. 40(5), 1605-1631 (2002) · Zbl 1037.65120
[37] Mardal, K.A., Winther, R.: An observation on Korn’s inequality for nonconforming finite element methods. Math. Comput. 75(253), 1-6 (2006) · Zbl 1086.65112
[38] Mu, L., Wang, J., Ye, X.: A weak Galerkin finite element method with polynomial reduction. J. Comput. Appl. Math. 285(C), 45-58 (2015) · Zbl 1315.65099
[39] Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element methods on polytopal meshes. Int. J. Numer. Anal. Model. 12(1), 31-53 (2015) · Zbl 1332.65172
[40] Scott, L.R., Vogelius, M.: Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO-Modélisation Mathématique et Analyse Numérique 19(1), 111-143 (1985) · Zbl 0608.65013
[41] Shi, Z., Ming, W.: Finite Element Methods. Science Press, Beijing (2013)
[42] Talischi, C., Paulino, G.H., Pereira, A., Menezes, I.F.M.: PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct. Multidisc. Optim. 45, 309-328 (2012) · Zbl 1274.74401
[43] Talischi, C., Pereira, A., Paulino, G.H., Menezes, I.F.M., Carvalho., M.S.: Polygonal finite elements for incompressible fluid flow. Int. J. Numer. Methods Fluids 74(2), 134-151 (2014) · Zbl 1455.76095
[44] Vacca, G.: An \[H^11\]-conforming virtual element for Darcy and Brinkman equations. Math. Models Methods Appl. Sci. 28(1), 159-194 (2018) · Zbl 1457.65218
[45] Wang, C., Wang, J., Wang, R., Zhang, R.: A locking-free weak Galerkin finite element method for elasticity problems in the primal formulation. J. Comput. Appl. Math. 307, 346-366 (2016) · Zbl 1338.74104
[46] Wang, J., Ye, X.: New finite element methods in computational fluid dynamics by H(div) elements. SIAM J. Numer. Anal. 45(3), 1269-1286 (2007) · Zbl 1138.76049
[47] Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241, 103-115 (2013) · Zbl 1261.65121
[48] Wang, J., Ye, X.: A weak Galerkin finite element method for the Stokes equations. Adv. Comput. Math. 42(1), 155-174 (2016) · Zbl 1382.76178
[49] Xie, X., Xu, J., Xue, G.: Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models. J. Comput. Math. 26(3), 437-455 (2008) · Zbl 1174.76013
[50] Zhang, S.: Divergence-free finite elements on tetrahedral grids for \[k\ge 6\] k≥6. Math. Comput. 80(274), 669-695 (2011) · Zbl 1410.76204
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.