By Bertrand Mercier

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**Example text**

U,v c SN (i) also called "pseudo-spectral" method. 3) _ = (~xu , av) N = (~ ' av (au, ~x)N Pc(aV))N - (u,Pc( a ~av)] x JN Pc(aV) ) _ (u,Pc( a ~v = -(u,Lcv). 1) yields a system which admits a unique solution u C s C0(0,T ; SN). 1: true for L and If a is constant, L C LN, we deduce that and L uC = u N coincide on SN; as is also and with it the equivalence of the Galerkin and collocation methods for this case. 1), then I-T llu(t) -Uc(t)ll 0 < C(I+N 2) 2 llu011r, for 0 ~ t ~< T. 1), and setting WN ~ ~ - uC ~WN az ~t + LcW N = (Lc-L)u N + ~ + Lz.

8), we have for LC is antisymmetric. u,v c SN (i) also called "pseudo-spectral" method. 3) _ = (~xu , av) N = (~ ' av (au, ~x)N Pc(aV))N - (u,Pc( a ~av)] x JN Pc(aV) ) _ (u,Pc( a ~v = -(u,Lcv). 1) yields a system which admits a unique solution u C s C0(0,T ; SN). 1: true for L and If a is constant, L C LN, we deduce that and L uC = u N coincide on SN; as is also and with it the equivalence of the Galerkin and collocation methods for this case. 1), then I-T llu(t) -Uc(t)ll 0 < C(I+N 2) 2 llu011r, for 0 ~ t ~< T.

2, with exactly the same results. 2: Let quantity N + = ¢ Estimate in the norm of Sobolev spaces of negative indices. be a given function sufficiently regular; we will show that the (¢, UN(t) - u(t)) even if u(t) converges "sufficiently rapidly" to zero as is not regular. For that purpose, we introduce the solution aW * t~--+ L W = O, w(o) where L* (= -L) Let of the adjoint problem t ~ 0 = ¢ is the adjoint of WN(t) ~ S N W L. be the solution of the approximate adjoint problem aW N , (~--~--+ L WN,VN) = 0, for all v N ~ SN (WN(O) - ¢,VN) = 0, for all v N e SN.