Multilevel Additive Schwarz Methods

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Multilevel Additive Schwarz Methods
Xuejun Zhang
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• On each level, associated with {Cl[}-J.^, there exists a partition of unity {5'} satis- fying J2^', = 1, wiih e', 6 ^o(^') n C°(Q|), 0 < e[ < l and |Ve. '| < C//i, _i. i The first property is very natural; it simply says that the restriction of the triangulation T' to subdomain Cl\ defines a triangulation for Cl'- and the finite element problem on fi' is well defined. The second condition is used to establish the upper bound of the spectrum of the additive Schwarz operator. The last condition is used for the lower bound of the spectrum. One way of constructing subdomains {(l'i}iJ, i, l = 2, •••, !. , with the above properties is described in Dryja and Widlund [5, 6]. Each triangle r, '~*, t = 1, • • •, iV/, / = 2, • • •, X, is extended to a larger region f'-~^ so that ch'~^ < dist(5f/~\5r/~^) < Ch'~^, aligning 5f, '"^ with the boundaries of level / triangles. We cut off the part of f/~^ that is outside il. We use f, '~^ as the subdomains Ct[. Another way of constructing {tl\} is given in section 4.

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