By Vivette Girault

The contents of this book were taught on the college Pierre & Marie Curie as a graduate direction in numerical research in the course of the educational 12 months 1977-78. within the previous couple of years, many engineers and mathematicians have centred their efforts at the numerical resolution of the Navier-Stokes equations by way of finite aspect equipment. the aim of this sequence of lectures is to supply a reasonably entire remedy of the newest mathematical advancements in that box. it's not meant to provide an exhaustive remedy of all finite aspect equipment to be had for fixing the Navier-Stokes equations. yet in its place, it areas a very good emphasis at the finite aspect tools of combined kind which play a primary half these days in numerical hydrodynamics. for this reason, those lecture notes can be seen as an creation to the combined finite point concept .We have attempted up to attainable to make this article self-contained. during this recognize, now we have recalled a couple of theoretical effects at the natural mathematical point of the Navier-Stokes challenge and we now have usually mentioned the new publication by way of R. Temam. The reader will locate during this reference additional mathematical fabric.

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**Example text**

4. u in If g'~ do = o and • {~E (HI(Cl»n ; div ~ = O} , o L2 (Q) = {q E L2 (Q) ; (q,l) = O} = o and we denote by J the orthogonal complement of V in scalar product (gr;d ;i, gr;d ~). The divergence operator is an isomorphism from Proof. 5 such that div ;i g ~ E (HI(Cl»n • By Green's formula o J onto 34 I -+--+- r V"V do = 0 • 2 Thus div E £«HI(Q}}n ; L (Q)}. ; (HI(Q»n o in v = Ker(div}, it suffices to show that div maps L (Q). Since onto L2 (Q). For this, let (HI(Q»n o function 0 2 on to e o -+- such that H2 (Q) in div v = q -+ grad eE 2 L (Q) ; we seek 0 ~ Q is bounded, there exists some As such that q ll8 = We set be a function of q o (HI (Q»n in Q Then • -+- d1V vI M I ~I q moreover, by Green's formula I ~I"~ r -+- Also Yov l E (H 1/2 there exists Then do = div Q dx = I q dx Q =0 .

2) (ii) ~ (iii). j),. ; defined by IIvEX. and it is easy to check that the correspondence g .......... ;) I o V • This permits to identify onto As a consequence, statements (ii) and (iii) are equivalent. 9) is usually called an " inf-sup " condition. 1. Let us make the following hypotheses : (i) There exists a constant such that a > 0 VvEV. 9). Then problem (P) has a unique solution unique A in M such that the pair (~,x) Moreover, the mapping ~ in V(x) and there exists a is the unique solution of problem (Q).

The proof for the three-dimensional case can be found in Temam [44 l. 3. OSi lion • of In solving the Stokes problem, we shall deal with divergence-free functions of (H1(Q»n. 5. 'J do o. such that = -+g r on PRX:>F. For the sake of simplicity, we assume that n =2 and that r is suf- ficiently smooth. The proof for the general case can be found in Temam [44 1• in 2 H (Q) 2i. dT Indeed, if we set = 0 ~ = c~rl and 1jJ , then Thus r. It suffices to find -+-+ E r . ; U'T u 2i. 22). ; E and H1/ 2 (r) , there exists a function on r .