Multiple Shooting and Time Domain Decomposition Methods by Thomas Carraro Michael Geiger Stefan Körkel & Rolf Rannacher

Author:Thomas Carraro, Michael Geiger, Stefan Körkel & Rolf Rannacher
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Then we have the following result that will be helpful to estimate for practical applications.

Lemma 4

Let Assumptions 2 and 3 hold and assume that (21) is satisfied with an imbedding operator . Then in Lemma 3 can be estimated by

(22)

Proof

Let be arbitrary. Then (A ∗ + M ∗ I ∗ C ∗)z ∈ ran(M ∗) and thus

Hence, we obtain

 □

2.2 Application to Parabolic Control Problems

We consider a parabolic state equation of the form

(23)

where , , , and y 0 ∈ L 2(Ω).

We set V = H 0 1(Ω), and Y: = W(0, T). Let Z = Z 1 × Z 2: = L 2(0, T; V ) × L 2(Ω) and assume that b ∈ Z 1 ∗ and . We work with the usual weak solutions and define the operator by

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