By Franco Brezzi, Michel Fortin

Research on non-standard finite point tools is evolving quickly and during this textual content Brezzi and Fortin provide a common framework within which the advance is occurring. The presentation is outfitted round a couple of vintage examples: Dirichlet's challenge, Stokes challenge, Linear elasticity. The authors offer with this booklet an research of the tools to be able to comprehend their homes as completely as possible.

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**Additional info for Mixed and Hybrid Finite Element Methods**

**Sample text**

3 for the Dirichlet problem. 15) on p. It is therefore possible to suppose div p + f = 0, as this can be attained by modifications to p that are internal to I~ (that is, not modifying p . 10). 16) f . 9. :i V ds = J8K . 17) . sup 1. K. t inf dlVf+J=o v ;EHl(K;)/lR i =l div 9. ' grad v dx, 19@dvi I2dx_fq. grad Vi dx}. JK;- -. 17) we evidently get, setting p . 18) g@d U i - -. ), where P is the projection operator in (L2(K;))2 on g@d(H1(K;)) . 20) sUI? )1 2 dx. 35). 20) is unnecessary. 14) does not hold.

J1 In - tr 'L- dx {{JK, div ~ . 1! 32) L { { divg. ;=1 JK, Q dx - {O"nn Q' J8K. 11 dS} Formally this is well defined for!!. chosen with while 1! 11 is continuous. Then theterm 1! 11 dS} = 0, + { t· '!!. dx = In Unt 0, V~, VQ. 32) can be read as div!!. + f = 0 in the sense of distributions. 32) for A +00 (that is, tr!!. , the case of an incompressible material. As we shall see, our main problem will then be to preserve symmetry in the discretized problem. a = = Up to now we considered a purely formal problem.

1 Mixed and Hybrid Finite Element Methods 45 Let us first consider a special case. 37) 31' > 0 such that c(q, q) ~ l' IIqll~, 'Vq E Q ~ a IIvll~, 'Vv E V. 38) 3a > 0 such that a(v,v) Then we have the following proposition. 36) has a unique solution (u, p). 39) a l' 2 21212 "2llullv + "2l1pllQ ::; 2allflivI + 21'lIgIl Q I. The proof is elementary. 39) is unsatisfactory. 40) c(p, q) = '\((p, q»Q, ,\ 0, ~ and we would like to get estimates that provide uniform bounds on the solution for ,\ small (say 0 ::; ,\ ::; 1).