Quasilinear Parabolic and Elliptic Differential Equations 3

The scalar product in Û{Q) is {v, w)= j w^ i; dx. Let

H^ { Q ) = {veHHQ)\v{x) = 0, xedQ}.

0

We extend ( , ) to give the duality between H^{Q) and its dual space H~^{Q), which is equipped with the operator norm, i.e.

\\v\U=s^xp { ( v , ф ) \фeHH^l\\Ф\\l = Ц^

0

Let $f^, depending on a "step size" parameter /г be a subspace of H^ (O), consisting of certain piecewise polynomials of degree r.

The family of subspaces {$y^} is assumed to satisfy the following asymptotic approximation assumption,

( C1 ) There exists a natural number r and a constant С such that if2^s^r+l and veH'{Q)nHHQ)

M\\v~x\\juCh^ - ^vl .

A variational formulation of (LI) is

{ u , , v ) + {Ä{u, Vu), Vv) + {f{u, Vu),v) = 0 \fvEHHQl (2.1)

du where щ=-—-, and the semidiscrete Galerkin approximation is now defined as dt

the solution U of

( Ц , 7) + (Л([7, VUl VV) + {fiU, VUl V) = 0 УУе$,, (2.2)

U { \0 ) will be defined later on. Let

a { u , w; v) = {A{u, Vu)-Ä{w, Fw), Vv) + {fiu, Vu)-f(w, Vw% v), (2.3)

0

u , w,veH^ (O), where apparently the functional a{u,w; v) is linear in v. In [1] it is shown that

( CZ ) a{u,w;u-w)^p\\V{u~w)f-Po\\u-wf, u,weH^{Q), where

p = inf < smallest eigenvalue of -^r- (w, F w) > > 0 w,FweKxK" [^ dç )

and the Gârding constant

Po = sup ^-div -™-w, Fw+-xV(w, Fw~-^(w, ^ w,Fw (2 IÔU 0^ л ou J