Basic principles of mixed Virtual Element Methods
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 (2014) no. 4, pp. 1227-1240.

The aim of this paper is to give a simple, introductory presentation of the extension of the Virtual Element Method to the discretization of H(div)-conforming vector fields (or, more generally, of (n - 1) - Cochains). As we shall see, the methods presented here can be seen as extensions of the so-called BDM family to deal with more general element geometries (such as polygons with an almost arbitrary geometry). For the sake of simplicity, we limit ourselves to the 2-dimensional case, with the aim of making the basic philosophy clear. However, we consider an arbitrary degree of accuracy k (the Virtual Element analogue of dealing with polynomials of arbitrary order in the Finite Element Framework).

DOI: 10.1051/m2an/2013138
Classification: 65N30, 65N12, 65N15, 76R50
Keywords: mixed formulations, virtual elements, polygonal meshes, polyhedral meshes
@article{M2AN_2014__48_4_1227_0,
     author = {Brezzi, F. and Falk, Richard S. and Donatella Marini, L.},
     title = {Basic principles of mixed {Virtual} {Element} {Methods}},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1227--1240},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {4},
     year = {2014},
     doi = {10.1051/m2an/2013138},
     mrnumber = {3264352},
     zbl = {1299.76130},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2013138/}
}
TY  - JOUR
AU  - Brezzi, F.
AU  - Falk, Richard S.
AU  - Donatella Marini, L.
TI  - Basic principles of mixed Virtual Element Methods
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2014
SP  - 1227
EP  - 1240
VL  - 48
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2013138/
DO  - 10.1051/m2an/2013138
LA  - en
ID  - M2AN_2014__48_4_1227_0
ER  - 
%0 Journal Article
%A Brezzi, F.
%A Falk, Richard S.
%A Donatella Marini, L.
%T Basic principles of mixed Virtual Element Methods
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2014
%P 1227-1240
%V 48
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2013138/
%R 10.1051/m2an/2013138
%G en
%F M2AN_2014__48_4_1227_0
Brezzi, F.; Falk, Richard S.; Donatella Marini, L. Basic principles of mixed Virtual Element Methods. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 (2014) no. 4, pp. 1227-1240. doi : 10.1051/m2an/2013138. http://www.numdam.org/articles/10.1051/m2an/2013138/

[1] 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. | MR

[2] D.N. Arnold, D. Boffi and R.S. Falk, Approximation by quadrilateral finite elements. Math. Comput. 71 (2002) 909-922. | MR | Zbl

[3] D.N. Arnold, D. Boffi and R.S. Falk, Quadrilateral H(div) finite elements. SIAM J. Numer. Anal. 42 (2005) 2429-2451. | MR | Zbl

[4] L. Beirão Da Veiga, F. Brezzi, A. Cangiani, L.D. Marini, G. Manzini and A. Russo, The basic principles of Virtual Elements Methods. Math. Models Methods Appl. Sci. 23 (2013) 199-214. | MR

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

[6] L. Beirão Da Veiga, F. Brezzi, L.D. Marini and A. Russo, Mixed Virtual Element Methods in three dimensions. In preparation.

[7] L. Beirão Da Veiga, K. Lipnikov and G. Manzini, Convergence analysis of the high-order mimetic finite difference method. Numer. Math. 113 (2009) 325-356. | MR | Zbl

[8] 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. | MR | Zbl

[9] L. Beirão Da Veiga and G. Manzini, A higher-order formulation of the Mimetic Finite Difference Method SIAM J. Sci. Comput. 31 (2008) 732-760. | MR | Zbl

[10] P. Bochev and J.M. Hyman, Principle of mimetic discretizations of differential operators, Compatible discretizations. In vol. 142 of Proc. of IMA hot topics workshop on compatible discretizations. Edited by D. Arnold, P. Bochev, R. Lehoucq, R. Nicolaides and M. Shashkov. Springer-Verlag (2006). | MR | Zbl

[11] D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications. Springer-Verlag, New York (2013). | MR | Zbl

[12] S.C. Brenner and R.L. Scott, The mathematical theory of finite element methods. In vol. 15 of Texts Appl. Math. Springer-Verlag, New York (2008). | MR | Zbl

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

[14] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). | MR | Zbl

[15] F. Brezzi, K. Lipnikov and M. Shashkov, Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Num. Anal. 43 (2005) 1872-1896. | MR | Zbl

[16] F. Brezzi, K. Lipnikov, M. Shashkov and V. Simoncini, A new discretization methodology for diffusion problems on generalized polyhedral meshes. Comput. Meth. Appl. Mech. Engrg. 196 (2007) 3682-3692. | MR | Zbl

[17] F. Brezzi, K. Lipnikov and V. Simoncini, A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15 (2005) 533-1553. | MR | Zbl

[18] F. Brezzi and L.D. Marini, Virtual elements for plate bending problems. Comput. Meth. Appl. Mech. Engrg. 253 (2013) 155-462. | MR | Zbl

[19] 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. | MR | Zbl

[20] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978). | MR | Zbl

[21] J. Douglas, Jr. and J.E. Roberts, Mixed finite element methods for second order elliptic problems. Math. Appl. Comput. 1 (1982) 91-103. | MR | Zbl

[22] 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. (M3AS) 20 (2010) 265-295. | MR | Zbl

[23] C.A. Felippa, Supernatural QUAD4: A template formulation Comput. Methods Appl. Mech. Engrg. 195 (2006) 5316-5342. | MR | Zbl

[24] T.-P. Fries and T. Belytschko, The extended/generalized finite element method: An overview of the method and its applications Int, J. Numer. Meth. Engng. 84 (2010) 253-304. | MR | Zbl

[25] A. Gain and G.H. Paulino, Phase-field based topology optimization with polygonal elements: a finite volume approach for the evolution equation. Struct. Multidiscip. Optim. (2012) 4632-7342. | MR | Zbl

[26] V. Gyrya and K. Lipnikov, High-order mimetic finite difference method for diffusion problems on polygonal meshes. J. Comput. Phys. 227 (2008) 8841-8854. | MR | Zbl

[27] J.M. Hyman and M. Shashkov, The orthogonal decomposition theorems for mimetic finite difference methods. SIAM J. Numer. Anal. 36 (1999) 788-818. | MR | Zbl

[28] Yu. Kuznetsov and S. Repin, New mixed finite element method on polygonal and polyhedral meshes. Russ. J. Numer. Anal. Math. Model. 18 (2003) 261-278. | MR | Zbl

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

[30] A. Tabarraei and N. Sukumar, Conforming polygonal finite elements. Int. J. Numer. Meth. Engrg. 61 (2004) 2045-2066. | MR | Zbl

[31] A. Tabarraei and N. Sukumar, Extended finite element method on polygonal and quadtree meshes. Comput. Methods Appl. Mech. Engrg. 197 (2007) 425-438. | MR | Zbl

[32] C. Talischi, G.H. Paulino and C.H. Le, Honeycomb Wachspress finite elements for structural topology optimization. Struct. Multidiscip. Optim. 37 (2009) 569-583. | MR | Zbl

[33] E. Wachspress, A rational Finite Element Basis. Academic Press, New York (1975). | MR | Zbl

Cited by Sources: