Some nonconforming finite elements for the plate bending problem

On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers

“…as an approximation of (2.21) by << non-conforming éléments» (see e.g. [17], [21], [30], [31]); therefore given Z ^ V, a(u, v) and ƒ € F', and given a closed subspace Z h C y 9 the existence of W, W h9 b(v 9 9) such that the hypotheses of corollary 1.1 are satisfied, constitutes some kind of « abstract patch-test » for the éléments of Z h (see also, in the case of the elasticity and plate bending problems, F. de Veubeke [15] ; also important in this contest are the papers by Raviart-Thomas [27] and Thomas [32]). formation that maps K on O £ and let P Vth P Wti the images of P v and P w (resp.)…”

“…If this is the case, then we get that the constant â h which appears in (2.20) is in fact independent of h. The vérifica-tion of Hl can, also in this case, be easily performed using (1.44), (1.47), (1.48), if we know the value of U and V on 8^. F. de Veubeke has shown (cfr« [15]) that in this case Z h is in fact a space of non conforming approximations of HQ(Q) and that we can find in this way all the classical non conforming éléments for the biharmonic problem (see [30], [31], [21]); for further details we refer to F. de Veubeke [15]; in a forthcoming paper we shall treat this case from a mathematical point of view.…”

“…The interest of these methods, and in particular of hybrid methods, has been increased by papers [15], [27], [32], in which the theory of « non conforming » (or « delinquent ») éléments (see e.g. [30], [21], etc.) is presented as a « particular case » (in some sense) of hybrid methods.…”

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“…For nonconforming elements, the basic mathematical theory has been studied in many works (see [8,16], [22]- [24], [33]). For quasi-conforming elements, detailed discussions can be found in [32,33].…”

“…In two dimensions, there are well-known nonconforming elements, including the elements named after Morley, Zienkiewicz, Adini, Bogner-Fox-Schmit, etc. (see [4,8,16,18]). There are some other ways of constructing elements, such as the quasi-conforming method [25,7].…”