By P. Ladeveze, J.T. Oden

Studying modelling, and specifically numerical versions, is changing into a vital and relevant query in smooth computational mechanics. numerous instruments, capable of quantify the standard of a version with reference to a different one taken because the reference, were derived. utilized to computational recommendations, those instruments result in new computational equipment that are referred to as "adaptive". the current e-book is anxious with outlining the state-of-the-art and the newest advances in either those very important areas.Papers are chosen from a Workshop (Cachan 17-19 September 1997) that's the 3rd of a chain dedicated to mistakes Estimators and Adaptivity in Computational Mechanics. The Cachan Workshop handled most recent advances in adaptive computational tools in mechanics and their affects on fixing engineering difficulties. It was once headquartered too on supplying solutions to basic questions reminiscent of: what's getting used or can be utilized at the moment to unravel engineering difficulties? What may be the kingdom of artwork within the 12 months 2000? What are the hot questions concerning mistakes estimators and their applications?"

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If ~p r O, let V h be such that ~Ph ~ O. Then, there exists a constant a, 0 < a < 1 such that: I~ - CAI, _< o-I,al~ 9 (25) P r o o f Since I~,hl, # 0, I~, - w~l, i~ strictly less than I~,1, by (24). T h e o r e m 2 Let ~p E V, ~Ph E ~h and ~ h satisfy the conditions of Lemma 5. Then, there exists a constant a, 0 < a < 1, such that: ~i -a 2 II'rr < I~,,1~ < I1~11.. (26) P r o o f The upper bound follows from equality (24). Indeed, I~1, _< I~1, -- II~TII.. 2) (27) I~o1~ < I~',,1~, from which the lower bound follows.

The errors in the velocity and pressure variables are driven by the residuals Tr and R~ in the momentum equation and the continuity equation respectively. The residuals represent the sources of error in the finite element approximations, and as such, are post-processed to provide meaningful error estimates in global "energy" norms. The evaluation of ~ is shown to be exact, local and cheap. The calculation of the error measure Tr is however more demanding. A new technique has been developed which provides accurate approximations of ~ ' through a global but inexpensive iterative process.

3, (1987), 243-250. [16] M. T. Oden, A unified approach to a-posteriori error estimation using element residual methods, Numer. , 65, (1993), 23-50. Z. Zhu, A posteriori error estimation- The relationship between different procedures, Com. Meth. Appl. Mech. Eng. C. L. Taylor, The Finite Element Method, IV Edition, Vol. I (1989), Vol. M. Chen, Superconvergence of finite element solutions and their derivatives, Numer. math. J. , 3, (1981), 118-125. [20] M. Zlamal, Superconvergence and reduced integration in the finite element method, Math.

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