A multigrid algorithm for the p-version of the virtual element method
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 337-364.

We present a multigrid algorithm for the solution of the linear systems of equations stemming from the p-version of the virtual element discretization of a two-dimensional Poisson problem. The sequence of coarse spaces are constructed decreasing progressively the polynomial approximation degree of the virtual element space, as in standard p-multigrid schemes. The construction of the interspace operators relies on auxiliary virtual element spaces, where it is possible to compute higher order polynomial projectors. We prove that the multigrid scheme is uniformly convergent, provided the number of smoothing steps is chosen sufficiently large. We also demonstrate that the resulting scheme provides a uniform preconditioner with respect to the number of degrees of freedom that can be employed to accelerate the convergence of classical Krylov-based iterative schemes. Numerical experiments validate the theoretical results.

DOI : 10.1051/m2an/2018007
Classification : 65N30, 65N55
Mots clés : Polygonal meshes, virtual element methods, p Galerkin methods, p multigrid
Antonietti, Paola F. 1 ; Mascotto, Lorenzo 1 ; Verani, Marco 1

1
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     title = {A multigrid algorithm for the p-version of the virtual element method},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {337--364},
     publisher = {EDP-Sciences},
     volume = {52},
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     doi = {10.1051/m2an/2018007},
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     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2018007/}
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Antonietti, Paola F.; Mascotto, Lorenzo; Verani, Marco. A multigrid algorithm for the p-version of the virtual element method. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 337-364. doi : 10.1051/m2an/2018007. http://archive.numdam.org/articles/10.1051/m2an/2018007/

[1] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. (1964). | Zbl

[2] R.A. Adams and J.J.F. Fournier, Sobolev Spaces. Academic Press (2003) Vol. 140. | MR | Zbl

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

[4] P.F. Antonietti, F. Brezzi and L.D. Marini, Bubble stabilization of discontinuous Galerkin methods. Comput. Methods Appl. Mech. Eng. 198 (2009) 1651–1659. | DOI | MR | Zbl

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

[6] 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

[7] P.F. Antonietti, M. Sarti and M. Verani, Multigrid algorithms for hp-discontinuous Galerkin discretizations of elliptic problems. SIAM J. Numer. Anal. 53 (2015) 598–618. | DOI | MR | Zbl

[8] P.F. Antonietti, L. Beirão Da Veiga, S. Scacchi and M. Verani, A C1 virtual element method for the Cahn-Hilliard equation with polygonal meshes. SIAM J. Numer. Anal. 54 (2016) 34–56. | DOI | MR | Zbl

[9] P.F. Antonietti, C. Facciolá, A. Russo and M. Verani, Discontinuous Galerkin approximation of flows in fractured porous media on polygonal and polyhedral meshes. MOX Report 55/2016 (2016).

[10] P.F. Antonietti, M. Sarti and M. Verani, Multigrid algorithms for high order discontinuous Galerkin methods. Lect. Notes Comput. Sci. Eng. 104 (2016) 3–13. | DOI | MR | Zbl

[11] P.F. Antonietti, M. Bruggi, S. Scacchi and M. Verani. On the virtual element method for topology optimization on polygonal meshes: a numerical study. Comput. Math. Appl. 74 (2017) 1091–1109. | DOI | MR | Zbl

[12] P.F. Antonietti, X. Hu, P. Houston, M. Sarti and M. Verani, Multigrid algorithms for hp-version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes. Calcolo 54 (2017) 1169–1198. | DOI | MR | Zbl

[13] P.F. Antonietti, G. Manzini and M. Verani, The fully nonconforming virtual element method for biharmonic problems. M3AS: Math. Models Methods Appl. Sci. 28 (2018) 387–407. | MR | Zbl

[14] B. Ayuso De Dios, K. Lipnikov and G. Manzini, The nonconforming virtual element method. ESAIM: M2AN 50 (2016) 879–904. | DOI | Numdam | MR | Zbl

[15] F. Bassi,L. Botti, A. Colombo, D.A. Di Pietro and P. Tesini, On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations. J. Comput. Phys. 231 (2012) 45–65. | DOI | MR | Zbl

[16] F. Bassi, L. Botti, A. Colombo, F. Brezzi and G. Manzini, Agglomeration-based physical frame DG discretizations: an attempt to be mesh free. M3AS: Math. Model. Methods Appl. Sci. 24 (2014) 1495–1539. | MR | Zbl

[17] L. Beirão Da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. Marini and A. Russo, Basic principles of virtual element methods. M3AS: Math. Model. Methods Appl. Sci. 23 (2013) 199–214. | MR | Zbl

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

[19] L. Beirão Da Veiga, F. Brezzi, L.D. Marini and A. Russo, The Hitchhiker’s guide to the virtual element method. M3AS: Math. Model. Methods Appl. Sci. 24 (2014) 1541–1573. | MR | Zbl

[20] L. Beirão Da Veiga, K. Lipnikov and G. Manzini, The Mimetic Finite Difference Method for Elliptic Problems. Vol. 11 of MS&A. Model. Simul. Appl. Springer, Cham (2014). | MR | Zbl

[21] 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. Eng. 295 (2015) 327–346. | DOI | MR | Zbl

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

[23] L. Beirão Da Veiga, A. Chernov, L. Mascotto and A. Russo, Basic principles of hp virtual elements on quasiuniform meshes. M3AS: Math. Model. Methods Appl. Sci. 26 (2016) 1567–1598. | MR | Zbl

[24] L. Beirão Da Veiga, F. Brezzi, L. Marini and A. Russo, Virtual element method for general second order elliptic problems on polygonal meshes. M3AS: Math. Model. Methods Appl. Sci. 26 (2016) 729–750. | MR | Zbl

[25] L. Beirão Da Veiga, F. Brezzi, L.D. Marini and A. Russo, Mixed virtual element methods for general second order elliptic problems on polygonal meshes. ESAIM: M2AN 50 (2016) 727–747. | DOI | Numdam | MR | Zbl

[26] L. Beirão Da Veiga, F. Brezzi, L.D. Marini and A. RussoH (div) and H(curl)-conforming virtual element methods. Numer. Math. 133 (2016) 303–332. | MR | Zbl

[27] L. Beirão Da Veiga, F. Brezzi, L.D. Marini and A. Russo, Serendipity nodal VEM spaces. Comput. Fluids 141 (2016) 2–12. | DOI | MR | Zbl

[28] L. Beirão Da Veiga, A. Chernov, L. Mascotto and A. Russo, Exponential convergence of the hp virtual element method with corner singularity. Numer. Math. 138 (2018) 581–613. | DOI | MR | Zbl

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

[30] C. Bernardi and Y. Maday, Polynomial interpolation results in Sobolev spaces. J. Comput. Appl. Math. 43 (1992) 53–80. | DOI | MR | Zbl

[31] S. Bertoluzza, M. Pennacchio and D. Prada, BDDC and FETI-DP for the virtual element method. Calcolo 54 (2017) 1565–1593. | DOI | MR | Zbl

[32] J.H. Bramble, Multigrid Methods. CRC Press (1993) Vol. 294. | MR | Zbl

[33] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 3rd edn. Vol. 15 of Texts in Applied Mathematics. Springer-Verlag, New York (2008). | DOI | MR | Zbl

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

[35] 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

[36] 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

[37] A. Cangiani, E.H. Georgoulis and P. Houston, hp-version discontinuous Galerkin methods on polygonal and polyhedral meshes. M3AS: Math. Model. Methods Appl. Sci. 24 (2014) 2009–2041. | MR | Zbl

[38] A. Cangiani, Z. Dong, E. Georgoulis and P. Houston, hp-version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes. ESAIM: M2AN 50 (2016) 699–725. | DOI | Numdam | MR | Zbl

[39] A. Cangiani, V. Gyrya and G. Manzini, The nonconforming virtual element method for the Stokes equations. SIAM J. Numer. Anal. 54 (2016) 3411–3435. | DOI | MR | Zbl

[40] A. Cangiani, Z. Dong and E. Georgoulis. hp-version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes. SIAM J. Sci. Comput. 39 (2017) A1251–A1279. | DOI | MR | Zbl

[41] A. Cangiani, E.H. Georgoulis, T. Pryer and O.J. Sutton, A posteriori error estimates for the virtual element method. Numer. Math. 137 (2017) 857–893. | DOI | MR | Zbl

[42] B. Cockburn, B. Dong and J. Guzmán, A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comput. 77 (2008) 1887–1916. | DOI | MR | Zbl

[43] B. Cockburn, D.A. Di Pietro and A. Ern, Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods. ESAIM: M2AN 50 (2016) 635–650. | DOI | Numdam | MR | Zbl

[44] F. Dassi and L. Mascotto, Exploring high-order three dimensional virtual elements: bases and stabilizations. Comput. Math Appl. 75 (2018) 3379–3401. | DOI | MR | Zbl

[45] D.A. Di Pietro, A. Ern and S. Lemaire, An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Comput. Methods Appl. Math. 14 (2014) 461–472. | DOI | MR | Zbl

[46] J. Droniou, R. Eymard, T. Gallouet and R. Herbin, Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations. M3AS: Math. Model. Methods Appl. Sci. 23 (2013) 2395–2432. | MR | Zbl

[47] L.C. Evans, Partial Differential Equations. American Mathematical Society (2010). | MR | Zbl

[48] R. Eymard, C. Guichard and R. Herbin, Small-stencil 3D schemes for diffusive flows in porous media. ESAIM: M2AN 46 (2012) 265–290. | DOI | Numdam | MR | Zbl

[49] M. Frittelli and I. Sgura, Virtual element method for the Laplace Beltrami equation on surfaces. To appear in: ESAIM: M2AN DOI: (2017). | DOI | Numdam | MR | Zbl

[50] 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. Eng. 282 (2014) 132–160. | DOI | MR | Zbl

[51] J. Hyman, M. Shashkov and S. Steinberg, The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials. J. Comput. Phys. 132 (1997) 130–148. | DOI | MR | Zbl

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

[53] L. Mascotto, Ill-conditioning in the virtual element method: stabilizations and bases. To appear in: Numer. Methods Partial Differ. Equ. DOI: (2017). | DOI | MR | Zbl

[54] D. Mora, G. Rivera and R. Rodríguez, A virtual element method for the Steklov eigenvalue problem. M3AS: Math. Model. Methods Appl. Sci. 25 (2015) 1421–1445. | MR | Zbl

[55] I. Perugia, P. Pietra and A. Russo, A plane wave virtual element method for the Helmholtz problem. ESAIM: M2AN 50 (2016) 783–808. | DOI | Numdam | MR | Zbl

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

[57] C. Schwab, p-and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Clarendon Press, Oxford (1998). | MR | Zbl

[58] N. Sukumar and A. Tabarraei, Conforming polygonal finite elements. Int. J. Numer. Methods Eng. 61 (2014) 2045–2066. | DOI | MR | Zbl

[59] A. Tabarraei and N. Sukumar, Extended finite element method on polygonal and quadtree meshes. Comput. Methods Appl. Mech. Eng. 197 (2008) 425–438. | DOI | MR | Zbl

[60] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces. Springer Science & Business Media Vol. 3 (2007). | MR | Zbl

[61] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. North-Holland (1978). | MR | Zbl

[62] J. Zhao, S. Chen and B. Zhang, The nonconforming virtual element method for plate bending problems. M3AS: Math. Model. Methods Appl. Sci. 26 (2016) 1671–1687. | MR | Zbl

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