By Vidar Thomee

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the examining of this booklet wishes the data of a few easy thought on finite components approximation, and a little of PDE's thought. in spite of the fact that, this information is no longer indispensable.

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17) Fh (t)vI 2 ≤ Ch2q |vI |2q = Ch2q 2q −(q−s) = Ch t tλm ≤1 λqm (v, ϕm )2 tλm ≤1 (tλm )q−s λsm (v, ϕm )2 ≤ Ch2q t−(q−s) |v|2s . We also note that Fh (t)vII ≤ Chq t−q/2 vII . 48 3. 2 since then hr t−r/2 ≤ hq t−q/2 . Since vII 2 = tλm >1 (v, ϕm )2 ≤ ts ∞ m=1 λsm (v, ϕm )2 = ts |v|2s , we conclude Fh (t)vII ≤ Chq t−(q−s)/2 |v|s . 18) show our claim. ⊓ ⊔ We shall now brieﬂy describe an alternative way of deriving the above nonsmooth data error estimates in the case of the standard Galerkin method, in which the main technical device is the use of a dual backward inhomogeneous parabolic equation with vanishing ﬁnal data, and which avoids the use of the operators Th and T .

65) −∆u + u = f in Ω, with ∂u =0 ∂n on ∂Ω. In order to formulate this in variational form, we now multiply by ϕ ∈ H 1 , thus without requiring ϕ = 0 on ∂Ω, integrate over Ω, and use Green’s formula to obtain 22 1. The Standard Galerkin Method (∇u, ∇ϕ) + (u, ϕ) = (f, ϕ), ∀ϕ ∈ H 1 . 65) follows since ϕ is arbitrary. In particular, the boundary condition is now a consequence of the variational formulation, in contrast to our earlier discussion when the boundary condition was enforced by looking for a solution in H01 .

27) for t > 0, with u3 (0) = 0. We notice that f1 and f3 vanish for t ≤ t0 − δ and t ≥ t0 − 3δ/4, respectively. 27) with u1,h (0) = u3,h (0) = 0, u2,h (0) = Ph v, and 3 set ej = uj,h − uj . Since, by linearity, e = uh − u = j=1 ej , it suﬃces to estimate ej (t0 ), j = 1, 2, 3, by the right-hand side of the estimate claimed. Consider ﬁrst the error in u1 . 26) by diﬀerentiation and Dtl u1,h its discrete counterpart, with both these functions vanishing for small t, Dtl e1 (t0 ) ≤ Chr t0 0 Dtl+1 u1 r ds ≤ Chr t0 j≤l+1 t0 −δ Dtj u r ds.