Divergence free virtual elements for the stokes problem on polygonal meshes

Beirão da Veiga, Lourenco; Lovadina, Carlo; Vacca, Giuseppe

doi:10.1051/m2an/2016032

Beirão da Veiga, Lourenco ¹ ; Lovadina, Carlo ^2,³ ; Vacca, Giuseppe ⁴

¹ Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Via Roberto Cozzi 55, 20125 Milano, Italy.
² Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy.
³ IMATI-CNR, Via Ferrata 1, 27100 Pavia, Italy.
⁴ Dipartimento di Matematica, Università degli Studi di Bari, Via Edoardo Orabona 4, 70125 Bari, Italy.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 509-535.

Résumé

In the present paper we develop a new family of Virtual Elements for the Stokes problem on polygonal meshes. By a proper choice of the Virtual space of velocities and the associated degrees of freedom, we can guarantee that the final discrete velocity is pointwise divergence-free, and not only in a relaxed (projected) sense, as it happens for more standard elements. Moreover, we show that the discrete problem is immediately equivalent to a reduced problem with fewer degrees of freedom, thus yielding a very efficient scheme. We provide a rigorous error analysis of the method and several numerical tests, including a comparison with a different Virtual Element choice.

MR Zbl | 4 citations dans Numdam

DOI : 10.1051/m2an/2016032

Classification : 65N30, 65N12, 65N15, 76D07
Mots-clés : Virtual element method, polygonal meshes, Stokes problem, divergence free approximation

Affiliations des auteurs :

Beirão da Veiga, Lourenco ¹ ; Lovadina, Carlo ^{2, 3} ; Vacca, Giuseppe ⁴

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     pages = {509--535},
     publisher = {EDP-Sciences},
     volume = {51},
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Beirão da Veiga, Lourenco; Lovadina, Carlo; Vacca, Giuseppe. Divergence free virtual elements for the stokes problem on polygonal meshes. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 509-535. doi : 10.1051/m2an/2016032. http://www.numdam.org/articles/10.1051/m2an/2016032/

Bibliographie
Cité par

B. Ahmad, A. Alsaedi, F. Brezzi, L.D. Marini and A. Russo, Equivalent projectors for virtual element methods. Comput. Math. Appl. 66 (2013) 376–391. | DOI | MR | Zbl

P.F. Antonietti, S. Giani and P. Houston, $h p$ -version composite discontinuous Galerkin methods for elliptic problems on complicated domains. SIAM J. Sci. Comput. 35 (2013) A1417–A1439. | DOI | MR | Zbl

P.F. Antonietti, L. Beirão Da Veiga, D. 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 | MR | Zbl

L. Beirão Da Veiga and G. Manzini, A virtual element method with arbitrary regularity. IMA J. Numer. Anal. 34 (2014) 759–781. | DOI | MR | Zbl

L. Beirão Da Veiga, V. Gyrya, K. Lipnikov and G. Manzini, Mimetic finite difference method for the Stokes problem on polygonal meshes. J. Comput. Phys. 228 (2009) 7215–7232. | DOI | MR | Zbl

L. Beirão Da Veiga, K. Lipnikov and G. Manzini, Arbitrary-order nodal mimetic discretizations of elliptic problems on polygonal meshes. SIAM J. Numer. Anal. 49 (2011) 1737–1760. | DOI | MR | Zbl

L. Beirão Da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (2013) 199–214. | DOI | MR | Zbl

L. Beirão Da Veiga, F. Brezzi and L.D. Marini, Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51 (2013) 794–812. | DOI | MR | Zbl

L. Beirão Da Veiga, F. Brezzi, L.D. Marini and A. Russo, The Hitchhiker’s Guide to the Virtual Element Method. Math. Models Methods Appl. Sci. 24 (2014) 1541–1573. | DOI | MR | Zbl

L. Beirão da Veiga, K. Lipnikov and G. Manzini, The mimetic finite difference method for elliptic problems. Vol. 11 of MS&A. Modeling, Simulation and Applications. Springer (2014). | MR | Zbl

L. Beirão Da Veiga, F. Brezzi, L.D. Marini and A. Russo, $H (d i v)$ and $H (c u r l)$ -conforming VEM. Numer. Math. (2015) 1–30.

L. Beirão Da Veiga, C. 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 | MR | Zbl

L. Beirão Da Veiga, C. Lovadina and G. Vacca, Divergence free virtual elements for the Stokes problem on polygonal meshes. Preprint (2015). | arXiv | Numdam | MR

M.F. Benedetto, S. Berrone, S. Pieraccini and S. Scialò, The virtual element method for discrete fracture network simulations. Comput. Methods Appl. Mech. Engrg. 280 (2014) 135–156. | DOI | MR | Zbl

J.E. Bishop, A displacement-based finite element formulation for general polyhedra using harmonic shape functions. Int. J. Numer. Methods Engrg. 97 (2014) 1–31. | DOI | MR | Zbl

D. Boffi, F. Brezzi and M. Fortin, Mixed finite element methods and applications. Vol. 44 of Springer Series in Computational Mathematics. Springer, Heidelberg (2013). | MR | Zbl

J. Bonelle and A. Ern, Analysis of compatible discrete operator schemes for elliptic problems on polyhedral meshes. ESAIM: M2AN 48 (2014) 553–581. | DOI | Numdam | MR | Zbl

S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Vol. 15 of Texts in Applied Mathematics, 3rd edition. Springer, New York (2008). | MR | Zbl

F. Brezzi and L.D. Marini, Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Engrg. 253 (2013) 455–462. | DOI | MR | Zbl

F. Brezzi, K. Lipnikov and M. Shashkov, Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43 (2005) 1872–1896. | DOI | MR | Zbl

F. Brezzi, A. Buffa and K. Lipnikov, Mimetic finite differences for elliptic problems. ESAIM: M2AN 43 (2009) 277–295. | DOI | Numdam | MR | Zbl

F. Brezzi, R. S. Falk and L.D. Marini, Basic principles of mixed virtual element methods. ESAIM: M2AN 48 (2014) 1227–1240. | DOI | Numdam | MR | Zbl

A. Buffa, C. De Falco and G. Sangalli, Isogeometric analysis: Stable elements for the 2d stokes equation. Int. J. Numer. Methods Fluids 65 (2011) 1407–1422. | DOI | MR | Zbl

A. Cangiani, E.H. Georgoulis and P. Houston, hp-version discontinuous galerkin methods on polygonal and polyhedral meshes. Math. Mod. Meth. Appl. Sci. 24 (2014) 2009–2041. | DOI | MR | Zbl

J. Carrero, B. Cockburn and D. Schötzau, Hybridized globally divergence-free LDG methods. I. The Stokes problem. Math. Comp. 75 (2006) 533–563. | DOI | MR | Zbl

P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77–84. | Numdam | MR | Zbl

B. Cockburn, G. Kanschat, D. Schötzau and C. Schwab, Local discontinuous Galerkin methods for the Stokes system. SIAM J. Numer. Anal. 40 (2002) 319–343. | DOI | MR | Zbl

B. Cockburn, D. Schötzau and J. Wang, Discontinuous Galerkin methods for incompressible elastic materials. Comput. Methods Appl. Mech. Engrg. 195 (2006) 3184–3204. | DOI | MR | Zbl

B. Cockburn, G. Kanschat and D. Schötzau, An equal-order DG method for the incompressible Navier–Stokes equations. J. Sci. Comput. 40 (2009) 188–210. | DOI | MR | Zbl

M. Costabel and M. Dauge, On the Inequalities of Babuška–Aziz, Friedrichs and Horgan–Payne. Arch. Ration. Mech. Anal. 217 (2015) 873–898. | DOI | MR | Zbl

D. Di Pietro and A. Ern, A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Methods Appl. Mech. Engrg. 283 (2015) 1–21. | DOI | MR | Zbl

R.G. Durán, An elementary proof of the continuity from L₂⁰(Ω) to H₀¹(Ω)ⁿ of Bogovskii’s right inverse of the divergence. Rev. Un. Mat. Argentina 53 (2012) 59–78. | MR | Zbl

M. Fortin and F. Brezzi, Mixed and hybrid finite element methods. Springer-Verlag, New York (1991). | MR | Zbl

A.L. Gain, C. Talischi and G.H. Paulino, On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes. Comput. Methods Appl. Mech. Engrg. 282 (2014) 132–160. | DOI | MR | Zbl

J. Guzmán and M. Neilan, Conforming and divergence-free Stokes elements in three dimensions. IMA J. Numer. Anal. 34 (2014) 1489–1508. | DOI | MR | Zbl

J. Guzmán and M. Neilan, Conforming and divergence-free Stokes elements on general triangular meshes. Math. Comp. 83 (2014) 15–36. | DOI | MR | Zbl

Y. Jeon, E.-J. Park and D. Sheen, A hybridized finite element method for the Stokes problem. Comput. Math. Appl. 68 (2014) 2222–2232. | DOI | MR | Zbl

K. Lipnikov, D. Vassilev and I. Yotov, Discontinuous galerkin and mimetic finite difference methods for coupled stokes–darcy flows on polygonal and polyhedral grids. Numer. Math. 126 (2013) 321–360. | DOI | MR | Zbl

K. Lipnikov, G. Manzini and M. Shashkov, Mimetic finite difference method. J. Comput. Phys. 257 (2014) 1163–1227. | DOI | MR | Zbl

D. Mora, G. Rivera and R. Rodríguez, A virtual element method for the steklov eigenvalue problem. Math. Models Methods Appl. Sci. 25 (2015) 1421–1445. | DOI | MR | Zbl

S.E. Mousavi and N. Sukumar, Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons. Comput. Mech. 47 (2011) 535–554. | DOI | MR | Zbl

S. Natarajan, S. Bordas and D.R. Mahapatra, Numerical integration over arbitrary polygonal domains based on Schwarz–Christoffel conformal mapping. Int. J. Numer. Meth. Eng. 80 (2009) 103–134. | DOI | MR | Zbl

A. Rand, A. Gillette and C. Bajaj, Interpolation error estimates for mean value coordinates over convex polygons. Adv. Comput. Math. 39 (2013) 327–347. | DOI | MR | Zbl

S. Rjasanow and S. Weißer, Higher order bem-based fem on polygonal meshes. SIAM J. Numer. Anal. 50 (2012) 2357–2378. | DOI | MR | Zbl

N. Sukumar and A. Tabarraei, Conforming polygonal finite elements. Int. J. Numer. Meth. Eng. 61 (2004) 2045–2066. | DOI | MR | Zbl

C. Talischi and G.H. Paulino, Addressing integration error for polygonal finite elements through polynomial projections: a patch test connection. Math. Models Methods Appl. Sci. 24 (2014) 1701–1727. | DOI | MR | Zbl

C. Talischi, G.H. Paulino, A. Pereira and I.F.M. Menezes, Polygonal finite elements for topology optimization: a unifying paradigm. Int. J. Numer. Meth. Eng. 82 (2010) 671–698. | DOI | Zbl

C. Talischi, G.H. Paulino, A. Pereira and I.F.M. Menezes, Polymesher: a general-purpose mesh generator for polygonal elements written in matlab. Struct. Multidisc Optimiz. 45 (2012) 309–328. | DOI | MR | Zbl

G. Vacca and L. Beirão Da Veiga, Virtual element methods for parabolic problems on polygonal meshes. Numer. Methods Partial Differ. Eq. 31 (2015) 2110–2134. | DOI | MR | Zbl

M. Vohralik and B.I. Wohlmuth, Mixed finite element methods: implementation with one unknown per element, local flux expressions, positivity, polygonal meshes, and relations to other methods. Math. Models Methods Appl. Sci. 23 (2013) 803–838. | DOI | MR | Zbl

E.L. Wachspress, Barycentric coordinates for polytopes. Comput. Math. Appl. 61 (2011) 3319–3321. | DOI | MR | Zbl

J.A. Wright and R.W. Smith, An edge-based method for the incompressible navier-stokes equations on polygonal meshes. J. Comput. Phys. 169 (2001) 24–43. | DOI | Zbl

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