Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-16T17:38:45.841Z Has data issue: false hasContentIssue false
  • English
  • Français

Basic principles of mixed Virtual Element Methods

Published online by Cambridge University Press:  15 July 2014

F. Brezzi ,
Richard S. Falk  and
L. Donatella Marini
F. Brezzi
Affiliation:
IUSS-Pavia and IMATI-CNR, Via Ferrata 1, 27100 Pavia, Italy.. brezzi@imati.cnr.it KAU, Jeddah, Saudi Arabia.
Richard S. Falk
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA.; falk@math.rutgers.edu
L. Donatella Marini
Affiliation:
Dipartimento di Matematica, Università di Pavia, and IMATI-CNR, Via Ferrata 1, 27100 Pavia, Italy.; marini@imati.cnr.it
Get access

Abstract

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).

Keywords

Mixed formulationsvirtual elementspolygonal meshespolyhedral meshes
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 4 , July 2014 , pp. 1227 - 1240
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahmad, B., Alsaedi, A., Brezzi, F., Marini, L.D. and Russo, A., Equivalent projectors for virtual element methods, Comput. Math. Appl. 66 (2013) 376–391. Google Scholar
Arnold, D.N., Boffi, D. and Falk, R.S., Approximation by quadrilateral finite elements. Math. Comput. 71 (2002) 909–922. Google Scholar
Arnold, D.N., Boffi, D. and Falk, R.S., Quadrilateral H(div) finite elements. SIAM J. Numer. Anal. 42 (2005) 2429–2451. Google Scholar
Beirão da Veiga, L., Brezzi, F., Cangiani, A., Marini, L.D., Manzini, G. and Russo, A., The basic principles of Virtual Elements Methods. Math. Models Methods Appl. Sci. 23 (2013) 199–214. Google Scholar
Beirão da Veiga, L., Brezzi, F. and Marini, L.D., Virtual Elements for linear elasticity problems. SIAM J. Num. Anal. 51 (2013) 794–812. Google Scholar
L. Beirão da Veiga, F. Brezzi, L.D. Marini and A. Russo, Mixed Virtual Element Methods in three dimensions. In preparation.
Beirão da Veiga, L., Lipnikov, K. and Manzini, G., Convergence analysis of the high-order mimetic finite difference method. Numer. Math. 113 (2009) 325–356. Google Scholar
Beirão da Veiga, L., Lipnikov, K. and Manzini, G., Arbitrary-Order Nodal Mimetic Discretizations of Elliptic Problems on Polygonal Meshes. SIAM J. Numer. Anal. 49 (2011) 1737–1760. Google Scholar
Beirão da Veiga, L. and Manzini, G., A higher-order formulation of the Mimetic Finite Difference Method SIAM J. Sci. Comput. 31 (2008) 732–760. Google Scholar
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).
D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications. Springer-Verlag, New York (2013).
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).
Brezzi, F., Buffa, A. and Lipnikov, K., Mimetic finite differences for elliptic problems. ESAIM: M2AN 43 (2009) 277–295. Google Scholar
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991).
Brezzi, F., Lipnikov, K. and Shashkov, M., Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Num. Anal. 43 (2005) 1872–1896. Google Scholar
Brezzi, F., Lipnikov, K., Shashkov, M. and Simoncini, V., A new discretization methodology for diffusion problems on generalized polyhedral meshes. Comput. Meth. Appl. Mech. Engrg. 196 (2007) 3682–3692. Google Scholar
Brezzi, F., Lipnikov, K. and Simoncini, V., A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15 (2005) 533–1553. Google Scholar
Brezzi, F. and Marini, L.D., Virtual elements for plate bending problems. Comput. Meth. Appl. Mech. Engrg. 253 (2013) 155–462. Google Scholar
Cangiani, A., Manzini, G. and Russo, A., Convergence analysis of the mimetic finite difference method for elliptic problems. SIAM J. Numer. Anal. 47 (2009) 2612–2637. Google Scholar
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978).
Douglas, J. Jr. and Roberts, J.E., Mixed finite element methods for second order elliptic problems. Math. Appl. Comput. 1 (1982) 91–103. Google Scholar
Droniou, J., Eymard, R., Gallouët, T. and Herbin, R., 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. Google Scholar
Felippa, C.A., Supernatural QUAD4: A template formulation Comput. Methods Appl. Mech. Engrg. 195 (2006) 5316–5342. Google Scholar
Fries, T.-P. and Belytschko, T., The extended/generalized finite element method: An overview of the method and its applications Int, J. Numer. Meth. Engng. 84 (2010) 253–304. Google Scholar
Gain, A. and Paulino, G.H., Phase-field based topology optimization with polygonal elements: a finite volume approach for the evolution equation. Struct. Multidiscip. Optim. (2012) 4632–7342. Google Scholar
Gyrya, V. and Lipnikov, K., High-order mimetic finite difference method for diffusion problems on polygonal meshes. J. Comput. Phys. 227 (2008) 8841–8854. Google Scholar
Hyman, J.M. and Shashkov, M., The orthogonal decomposition theorems for mimetic finite difference methods. SIAM J. Numer. Anal. 36 (1999) 788–818. Google Scholar
Kuznetsov, Yu. and Repin, S., New mixed finite element method on polygonal and polyhedral meshes. Russ. J. Numer. Anal. Math. Model. 18 (2003) 261–278. Google Scholar
Rjasanow, S. and Weißer, S., Higher order BEM-based FEM on polygonal meshes. SIAM J. Numer. Anal. 50 (2012) 2357–2378. Google Scholar
Tabarraei, A. and Sukumar, N., Conforming polygonal finite elements. Int. J. Numer. Meth. Engrg. 61 (2004) 2045–2066. Google Scholar
Tabarraei, A. and Sukumar, N., Extended finite element method on polygonal and quadtree meshes. Comput. Methods Appl. Mech. Engrg. 197 (2007) 425–438. Google Scholar
Talischi, C., Paulino, G.H. and Le, C.H., Honeycomb Wachspress finite elements for structural topology optimization. Struct. Multidiscip. Optim. 37 (2009) 569–583. Google Scholar
E. Wachspress, A rational Finite Element Basis. Academic Press, New York (1975).