SIAM J. NUMER. ANAL. c 2013 Society for Industrial and Applied Mathematics Vol. 51, No. 2, pp. 794–812 VIRTUAL ELEMENTS FOR LINEAR ELASTICITY PROBLEMS ∗ L. BEIR ˜ AO DA VEIGA † , F. BREZZI ‡ , AND L. D. MARINI § Abstract. We discuss the application of virtual elements to linear elasticity problems, for both the compressible and the nearly incompressible case. Virtual elements are very close to mimetic finite differences (see, forlinear elasticity, [L. Beir˜ao da Veiga, M 2AN Math. Model. Numer. Anal., 44 (2010), pp. 231–250]) and in particular to higher order mimetic finite differences. As such, they share the good features of being able to represent in an exact way certain physical properties (conservation, incompressibility, etc.) and of being applicable in very general geometries. The advantage of virtual elements is the ductility that easily allows high order accuracy and high order continuity. Key words. mimetic finite differences, virtual elements, elasticity AMS subject classifications. 65N30, 65N12, 65G99, 76R99 DOI. 10.1137/120874746 1. Introduction. In recent times the mimetic finite difference approach has been successfully applied to a great variety of problems, from diffusion problems to electromagnetism, on fairly irregular decompositions, including polygons with rather weird shapes, polyhedra in three dimensions with curved faces, hanging nodes, and so on. (For a partial list of citations we refer to [29, 25, 17, 24, 26, 13, 15, 8, 27, 1, 14, 6, 18, 21, 32, 11, 2, 10, 16].) Their use was limited to low order approximations until very recently, when people started to extend the methodology to include higher order approximations to gain better accuracy in the numerical results. See, e.g., [22, 5, 4, 3]. The first results in this direction were very encouraging, and people started to look more closely to these extensions, analyzing advantages and limitations and mostly looking for the key properties that could make things easier. This gave rise to a new interpretation of mimetic finite differences (see, to start with, [16]) and to a subsequent new approach, much closer to finite elements, that we call the virtual element method (VEM). Other methods that extend the Finite Element philosophy to polygonal meshes can be found in [30, 31, 28]. The basic idea of the new method can be described, roughly speaking, as follows. We start as we do for the classical finite elements of Lagrange or Hermite type, with one difference: in each element K, together with the usual polynomials, say, S (in general, all the polynomials up to a given degree k), some additional functions are also considered (typically solutions of PDEs within the element K) in order to get unisolvence. If things are properly done (good choice of the functions and of the degrees of freedom), the local stiffness matrix AE can be computed exactly whenever one of the two entries is a polynomial of S , without solving the local PDE (virtual ∗ Received by the editors April 23, 2012; accepted for publication (in revised form) December 26, 2012; published electronically March 5, 2013. This work was supported by King Abdulaziz University under grant 38-3-1432/HiCi, KAU, and MIUR (Italian Ministery of University and Research) under project PRIN2008. http://www.siam.org/journals/sinum/51-2/87474.html † Dipartimento di Matematica, Universit` a di Milano Statale, Milano, Italy (lourenco.beirao@ unimi.it). ‡ IUSS, IMATI-CNR, 27100 Pavia, Italy, and King Abdulaziz University, Department of Mathe- matics, Jeddah 21589, Saudi Arabia ([email protected]). § Dipartimento di Matematica, Universit`a di Pavia, and IMATI-CNR,27100 Pavia, Italy (marini@ imati.cnr.it). 794
VIRTUAL ELEMENTS FOR LINEAR ELASTICITY PROBLEMS
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