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Magioladitis (talk | contribs) m Improved article to match Manual of Style, References after punctuation per WP:CITEFOOT and WP:PAIC using AWB (12151) |
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The GDM is then in this case a nonconforming method for the approximation of (2), which includes the nonconforming finite element method. Note that the reciprocal is not true, in the sense that the GDM framework includes methods such that the function <math>\nabla_D u</math> cannot be computed from the function <math>\Pi_D u</math>.
The following error estimate, inspired by G. Strang's second lemma,<ref>'''G. Strang.''' Variational crimes in the finite element method.'' In The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972)'', pages 689–710. Academic Press, New York, 1972.</ref> holds
:<math>\quad (4) \qquad \qquad \Vert \nabla \overline{u} - \nabla_D u_D\Vert_{L^2(\Omega)^d}
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=== Nonconforming finite element ===
On a mesh <math>T</math> which is a conforming set of simplices of <math>\mathbb{R}^d</math>, the nonconforming <math>P^1</math> finite elements are defined by the basis <math>(\psi_i)_{i\in I}</math> of the functions which are affine in any <math>K\in T</math>, and whose value at the centre of gravity of one given face of the mesh is 1 and 0 at all the others (these finite elements are used in [Crouzeix ''et al'']<ref>'''M. Crouzeix and P.-A. Raviart.''' Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 7(R-3):33–75, 1973.</ref> for the approximation of the Stokes and [[Navier-Stokes equations]]). Then the method enters the GDM framework with the same definition as in the case of the Galerkin method, except for the fact that <math>\nabla\psi_i</math> must be understood as the "broken gradient" of <math>\psi_i</math>, in the sense that it is the piecewise constant function equal in each simplex to the gradient of the affine function in the simplex.
=== Mixed finite element ===
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=== Mimetic finite difference method and nodal mimetic finite difference method ===
This family of methods is introduced by [Brezzi ''et al'']<ref>'''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(5):1872–1896, 2005.</ref> and completed in [Lipnikov ''et al''].<ref>'''K. Lipnikov, G. Manzini, and M. Shaskov.''' Mimetic finite difference method. J. Comput. Phys., 257-Part B:1163–1227, 2014.</ref> It allows the approximation of elliptic problems using a large class of polyhedral meshes. The proof that it enters the GDM framework is done in [Droniou ''et al''].<ref name=droniou />
==See also==
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