Domain Decomposition Methods in Optimal Control of Partial by John E. Lagnese, Günter Leugering

V. 3. Domain Decomposition Methods for Elliptic Problems We have rji Ri /-l = /-l, for all i, j and hence rji Ri /-l = rij R j /-l. 20h holds. 45) On the other hand, for V E V, Vi = riV E Vi, trlr V =: /-l E W, we have ri v- Ri /-l E Vio, since rji(r; V - Ri/-l) = trlrv - /-l = 0 on r.

4) Update: k+l = qji k+~ qji + Pk (rji uik - 7] k) . 3. , 1, l)T, (1,0, of, (0, -1, l)T for (3 = Pk. Pk It therefore appears optimal to choose Pk = (3/2. The limiting value is (3 = Pk. 4 cannot be analyzed from the spectrum and the eigenbasis of Mkl Nk alone. Indeed, a convergence analysis for (3 1= Pk is not known in this context of domain decompositions. The case (3 = P for all k will be precisely the one considered in this monograph. See Glowinski and Le Tallec [28] for a discussion of the convergence properties of this version of the saddle-point algorithm.

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