By Roger Temam (Eds.)
This monograph relies on study undertaken by way of the authors over the last ten years. the most a part of the paintings offers with homogenization difficulties in elasticity in addition to a few mathematical difficulties relating to composite and perforated elastic fabrics. This learn of methods in strongly non-homogeneous media brings forth plenty of only mathematical difficulties that are extremely important for functions. even though the equipment prompt care for desk bound difficulties, a few of them could be prolonged to non-stationary equations. apart from a few recognized proof from practical research and the idea of partial differential equations, all leads to this booklet are given exact mathematical proof.
It is anticipated that the implications and strategies provided during this booklet will advertise additional research of mathematical types for tactics in composite and perforated media, heat-transfer, power move through radiation, procedures of diffusion and filtration in porous media, and they will stimulate study in different difficulties of mathematical physics and the idea of partial differential equations.
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Extra info for Navier-Stokes Equations: Theory and Numerical Analysis
Sample text
1 is also valid with u and V replaced by U and V•. 3. 10). 1. 5) cannot be extended by continuity to the closure V off. 11) is not satisfied, V is a Hilbert space for the norm [un = Vlul 2 + lIul1 2 , which is not equivalent to the norm lIuli since we lose the Poincare inequality. In order to pose and solve a variational problem in the general case, let us introduce the space Y = the completion of f under the norm 11•11. 32) VC Y. 3. Y C {u E La (n): DiU E L2 (n), 1 ~ l ~ n} with ex = 2n/(n-2) if n ~ 3, and YC {uELToc(n): for n = 2.
2. 12). 2) and Q is a linear continuous form on W. Let u denote the unique solution in W of a(u, v) = (Q,V) , ’Iv E W. ,) on WhxWh which is coercive and which, more precisely, satisfies Discretization of the Stokes equations Ch. 5) in which 1I•II*h stands for the norm in Wh, and in which {3 is indepen› dent of h. 3). A general theorem on the convergence of the approximate solutions uh to the exact solution will be given after defining precisely the mann› er in which the forms ah and Qh are consistent with the forms a and Q.
25) which makes it a Hilbert space. The functions Uh and 8iuh’ I ~ i ~ n, have compact supports in n, by the definition of Wh and the set nl. Hence they will be considered as vector functions defined on n or on \Rn . 26) The norm of Ph is exactly one, and they are stable. 27) which completely defines rhu E Wh . 2. The preceding ex ternal approximation of stable and convergent. Hb (n) is Proof. The approximation is stable since the prolongation operators are stable. 6. 1. 28) 52 The steady-state Stokes equations Ch.