On the numerical solution of plate bending problems by hybrid methods

Superconvergence of the gradient of finite element solutions

“…is a polynomial of degree n-\-1 in each variable, hence i; must vanish identically which proves the unisolvability of the Lagrange interpolation problem v {â 3 ) = v 3 , J = l, ...,(n + l) 2 . Let Njil,!, ^2) e ô(") ^e basic fonctions, i.e.…”

“…Now any polynomial v from Q(n) is uniquely determined by its values v 3 at âj. Let namely v{â 3 ) = 0, 7 = 1, ..., (n +1) 2 . The function t5(^i, s ( ) is a polynomial of degree not greater than nandit vanishesfor ^i=s fc , k = 0, .…”

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

“…through F t . We consider now the spaces for every décomposition, then the constant a h which appears in (2.12), is in fact independent of h. We remark that in practice, since the value of 9 in K is depending only on the values of 9 and 7-on dK, P w will be chosen as a on space of biharmonic functions such that 9 and ~-are polynominals of assigned valid whenevever Ü is sufficiently smooth and v ljXX + 2v 2fXy + t; 3 yy = 0 on K, For further details we refer to Brezzi [5] and Brezzi-Marini [8].…”

A hybrid finite element method to compute the free vibration frequencies of a clamped plate

“…Note that V h (l) ^ F(^f t ; 1). According to Brezzi-Marini [5], we choose the parameters m, r, s satisfying m ^ max (r -2, 5 -1) (unless r -2 = s -1 and s is even, in which case m ^ s), so that the following estimate holds In particular, for any g e £f we can uniquely solve the following problems, which approximate respectively (9), (10), (11) :…”

“…[15]) and analyzed by Brezzi [4] and Brezzi-Marini [5], essentially involves the tensors « selfequilibrating » over each element ; on the contrary the natural space for the dynamical case is some orthogonal complement to the tensors selfequilibrating over the whole domain, in that way we can reduce our problem to a spectral problem for a compact selfadjoint operator T, whose finite dimensional approximation is however not of inner type. Convergence and error estimâtes are then derived by a Rayleigh-Ritz principle ; since we cannot prove the uniform convergence of the approximate operators T h to T, we check an approximability property over suitable finite dimensional spaces, similar to Property P 2 by Rappaz [17] (actually, our approach is analogous to the one he uses for a second order problem ; see also Rappaz [16], and Anselone [1] whose notion of " collective compactness ", however, does not apply in the present case).…”