Quasilinear Parabolic and Elliptic Differential Equations 5

The first term on the right hand side is estimated by

\ { r , „ r ) \u\\ri , \u\\riuc\Mi , + ^\\vi^f ,

where we have used Friedrichs' inequality

||t ; ||^C|||7t ; || , veHHQ).

If we choose Z so that the second term is zero, which is the case if Z is the Galerkin solution for each t > 0 of the nonhnear elliptic problem

- div ^(t;, Vv)+f{v, Pi;)= -div(^(M, Vu)+f{u, Vu)

we will likely come into difficulties when estimating ||/;J_i. We have (see e.g.

[ 11 ] )

а { и , Z; V)= j IVV^, K] j/(«, Vu) \^^"~'^] dx Q L M —z J

where a is defined in (2.3), w = м + s (Z — w),

j / ( co , V(d) =

SA ^ ^ , dA ^ ^ -

— - ( со , Po)) -—-{(o^Vcd)

dç ou

L OÇ OU

and s/' is the Fréchet derivate. Thus we will instead let Z be the Galerkin solution of the corresponding hnearized problem, i.e. Z = Z(", t) is then for each t>0 the elliptic projection of u — u{',t) defined by the Hnearized operator ja/. We thus let Z be defined by the bilinear form

b { u , Vu; rj, V)= f [PF^ F] j/(w. Vu) \^Ц dx = 0 VFe^^, (3.3)

ß L ^J

where rj = u — Z.

For b we have (see the previous section and (C2))

b { u , Vu ; V , V ) ^ô\\Vv\\\ ^Ve$t,

and it is well-known how to derive asymptotic error estimates for such problems. The crucial trick is to use an L2-lifting,

INIIo^C / ill^il , , which follows from the elliptic regularity

ll^lli^Cll^illo of ^*{u)\l/--rj, xGß, ф=0, xedQ,

( 3 . 4 )