Residual a posteriori error estimation for the Virtual Element Method for elliptic problems

Beirão da Veiga, L.; Manzini, G.

doi:10.1051/m2an/2014047

Beirão da Veiga, L.¹ ; Manzini, G.^2,³

¹ Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, via Saldini 50, 20133 Milano, Italy
² Applied Mathematics and Plasma Physics Group, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
³ IMATI del CNR, via Ferrata 1, 27100 Pavia, Italy

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 2, pp. 577-599.

Résumé

A posteriori error estimation and adaptivity are very useful in the context of the virtual element and mimetic discretization methods due to the flexibility of the meshes to which these numerical schemes can be applied. Nevertheless, developing error estimators for virtual and mimetic methods is not a straightforward task due to the lack of knowledge of the basis functions. In the new virtual element setting, we develop a residual based a posteriori error estimator for the Poisson problem with (piecewise) constant coefficients, that is proven to be reliable and efficient. We moreover show the numerical performance of the proposed estimator when it is combined with an adaptive strategy for the mesh refinement.

Reçu le : 2013-11-03

MR Zbl

DOI : 10.1051/m2an/2014047

Classification : 65N30
Mots clés : A posteriori error estimation, virtual element method, polygonal mesh, high-order scheme

Affiliations des auteurs :

Beirão da Veiga, L. ¹ ; Manzini, G. ^{2, 3}

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     title = {Residual a posteriori error estimation for the {Virtual} {Element} {Method} for elliptic problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
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     publisher = {EDP-Sciences},
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Beirão da Veiga, L.; Manzini, G. Residual a posteriori error estimation for the Virtual Element Method for elliptic problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 2, pp. 577-599. doi : 10.1051/m2an/2014047. http://www.numdam.org/articles/10.1051/m2an/2014047/

Bibliographie
Cité par

B. Ahmed, 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

M. Ainsworth and J.T. Oden, A posteriori error estimation in finite element analysis. Comput. Methods Appl. Mech. Eng. 142 (1997) 1–88. | DOI | MR | Zbl

P.F. Antonietti, L. Beirão Da Veiga, C. Lovadina and M. Verani, Hierarchical a posteriori error estimators for the mimetic discretization of elliptic problems. SIAM J. Numer. Anal. 51 (2013) 654–675. | DOI | MR | Zbl

I. Babuska and W.C. Rheinboldt, Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15 (1978) 736–754. | DOI | MR | Zbl

R.E. Bank and E. Weiser, Some a posteriori error estimators for elliptic partial differential equations. Math. Comput. 44 (1985) 283–301. | DOI | MR | Zbl

L. Beirão Da Veiga, A residual based error estimator for the mimetic finite difference method. Numer. Math. 108 (2008) 387–406. | DOI | MR | Zbl

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 | MR | Zbl

L. Beirão Da Veiga and G. Manzini, An a posteriori error estimator for the mimetic finite difference approximation of elliptic problems. Int. J. Numer. Meth. Eng. 76 (2008) 1696–1723. | 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, 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) 119–214. | DOI | MR | Zbl

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

L. Beirão da Veiga, K. Lipnikov and G. Manzini, The Mimetic Finite Difference Method. In vol. 11 of Model. Simul. Appl., 1st edition. Springer-Verlag, New York (2013).

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

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

D. Braess, Finite elements. Theory, fast solvers, and applications in elasticity theory, 3rd edition. Cambridge University Press (2007). | 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 and L.D. Marini, Virtual element method for plate bending problems. Comput. Methods Appl. Mech. Eng. 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

S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, Berlin/Heidelberg (1994). | MR | Zbl

A. Cangiani and G. Manzini, Flux reconstruction and pressure post-processing in mimetic finite difference methods. Comput. Methods Appl. Mech. Eng. 197 (2008) 933–945. | DOI | MR | Zbl

A. Cangiani, G. Manzini and A. Russo, Convergence analysis of the mimetic finite difference method for elliptic problems. SIAM J. Numer. Anal. 47 (2009) 2612–2637. | DOI | MR | Zbl

P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978). | MR | Zbl

W. Dorfler and R.H. Nochetto, Small data oscillation implies the saturation assumption. Numer. Math. 91 (2002) 1–12. | DOI | MR | Zbl

J. Droniou, R. Eymard, T. Gallouët and R. Herbin, A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. Math. Models Methods Appl. Sci. 20 (2010) 265–295. | DOI | MR | Zbl

P. Grisvard, Elliptic problems in nonsmooth domains. In vol. 24 of Monogr. Stud. Math. Pitman, Boston (1985). | MR | Zbl

K. Lipnikov, G. Manzini and D. Svyatskiy, Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems. J. Comput. Phys. 230 (2011) 2620–2642. | DOI | MR | Zbl

S. 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

P. Neittaanmki and S. Repin, Error control and a posteriori estimates. Reliable methods for computer simulation. In vol. 33 of Stud. Math. Appl. Elsevier Science (2004). | MR | Zbl

S. Repin, A posteriori Estimates for Partial Differential Equations. In vol. 4 of Radon Series on Computational and Applied Mathematics. De Gruyter, Berlin (2008). | MR | Zbl

R. Stenberg, A technique for analysing finite-element methods for viscous incompressible flow. Int. J. Numer. Meth. Fluids 11 (1990) 935–948. | DOI | MR | Zbl

N. Sukumar and E. Malsch, Recent advances in the construction of polygonal finite element interpolants. Arch. Comput. Methods Eng. 13 (2006) 129–163. | 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. Methods Eng. 82 (2010) 671–698. | DOI | Zbl

C. Talischi, A. Pereira, G.H. Paulino, I.F.M. Menezes and M.S. Carvalho, Polygonal finite elements for incompressible fluid flow. Int. J. Numer. Methods Fluids 74 (2014) 134–151. | DOI | MR | Zbl

R. Verfürth, A review of a posteriori error estimation and adaptive mesh refinement. Wiley and Teubner, Stuttgart (1996). | Zbl

M. Vohralik, Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients. J. Sci. Comput. 46 (2011) 397–438. | 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. Model. Methods Appl. Sci. 23 (2013) 803–838. | DOI | MR | Zbl

E. Wachspress, A rational Finite Element Basis. Academic Press (1975). | MR | Zbl

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