By Curtis F. Gerald Patrick O. Wheatley

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**Example text**

Ch21[vll , and S2)Ipt (s) ll < s2H (Th-T')utl) +s211 (Th-T)~ttll < Ch2s2(II~t(s)II + II~tt(s)II) < Ch21[vll Since R(t) = It (Th-T)~tds = [ (Th-T)~(s) ]0 - 0 It (T~-T')~ ds , 0 we also have IIR(s) II< Ch 2 sup II~(y)II <_ Ch211vll , y < s and the stability of the solution operators gives at once lle(s)ll

It is well known, and proved in essentially the same way as for the selfadjoint case, that ll(Th-T)fll + h[l(Th-T)fll I ! chs[ifiIs_2 for 2 < s < r, and it thus remains to prove the corresponding result for the time derivative. ) e = Wh-W. ) by differentiating A'(. )), V X C Sh , X C Sh, lletll ~ ! C{]]ell~+ inf l[~t-xllI} <__chS-~{llflls_2+ llwtlls} xES h j chs-1 {llf[Is_2 + [lw]I~}! chs-IIlfJ[s_2 for 2 < s < r. Here we have used the fact that IIwtlIs! Cilwlls, which follows since w t E H~(~) is the solution of the Dirichlet problem vm 6 H~(~) A m(w t,~) = -A'(w,m) .

Anal. 14, 218-241(1977). 3. V. Thom~e, Negative norm estimates and superconvergence parabolic problems. Math. Comput. 34, 93-113(]980). in Galerkin methods for 3. SMOOTH AND NON-SMOOTH DATA ERROR ESTIMATES FOR THE HOMOGENEOUS We shall begin by introducing scribing the regularity EQUATION. some function spaces which are convenient of the solution of the homogeneous in de- equation. Consider thus the initial boundary value problem for the homogeneous heat equation, (I) u t = Au where ~ in 2 x [0,~) , u = 0 on ~x u(x,0) = v(x) [0,~) , in ~, is a bounded domain in Rd with smooth boundary on 3~.