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By Artur Portela, A. Charafi

This e-book is an important device written for use because the fundamental textual content for an undergraduate or early postgraduate path in addition to a reference booklet for engineers and scientists who are looking to fast improve finite-element courses.

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We take the L2-inner product of the error equation (9) by since and u 2 ds)I/2 -- 0 uh V v C L 2, and observe that 29 For the proof of our final result of this chapter we introduce the solution operator Eh(t) of the initial value problem for the homogeneous semidiscrete parabolic equation, (11) ThUh,t+Uh = 0 Note that Eh(t) for t ~ 0 . is the semigroup on Sh generated by Ah = and that (11) is equivalent to (12) Uh, t = AhU h for t ~ 0 . In the following lemma we shall prove that this semigroup is uniformly bounded and analytic.

The proof of the lemma is now complete. We return to the initial-boundary value problem (I), (2) and begin our error analysis in L2 with the following simple result for the nonhomogeneous equation which is analogous to the result in the model problem treated earlier. Theorem I. Assume that (i) and (ii) hold and let tlVh-Vl] i Chrilvllr " Then we have for the error in the semidiscrete parabolic problem t Jluh(t)-u(t)JJ ! Chr{Hvllr + I llUtHrdS} for 0 Proof. With the above notation, set e(t) = ~h(t) -u(t) = e -Kt (Uh (t )-u (t)_) We have then the error equation Thet+e = -p ~_ (Th-T)A ~ .

0 It follows n o w by Lerm~a 4 that II~(t)tl! c sup (sII~tt I + t]~tl) ! c sup (sliplI + II~ll), s

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