Domain Decomposition And Iterative Refinement Methods for Mixed Finite Element D

Cover Domain Decomposition And Iterative Refinement Methods for Mixed Finite Element D
Domain Decomposition And Iterative Refinement Methods for Mixed Finite Element D
Tarek P Mathew
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. . , Vlj^. Thus, if we assign zero normal trace on each subdomain boundary 5^2, we would have compatible data to pose N parallel subproblems on fii, . . . , Q, i^ for local velocities and pressures {6uf^, Sp\) using the new right hand side, given by the residual, as follows: Find 5u\ e Xhi^i), Sp[ G y^(J7. ) such that a{Sul, v) + b{v, 8p'^) = -a{I^un, v), W e Xh{^i) [ HSllq) = Fiq)-b{I^UH, q), Vg G ^(a).
The local spaces (A'"h(r2, ), ^/^(n, )) are defined by Y, {n, )^QH{n)nL'{no/R- Since the l
...ocal problems have compatible boundary conditions, see (2. 17), they are well posed. Since the normal traces are zero on interface boundaries, they match; thus this composite velocity will be in A';, (fi). See Lemma 5 about composite functions. We form the composite velocity by adding up the subdomain velocities 6uf^. Let By construction, Uh, ! satisfies BhUhj — BhUh = F^. Thus Uh, DF = ^h — 'Uh, i, satisfies: j a{uh, DF, v) + b{v, Ph) = -a{uh, i, v), W e Xh{^) (o -iR) 1 b{uH. DF, q) = 0, Vgen(fi), ^ ^^ i.

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