Additive Schwarz Methods for Elliptic Finite Element Problems in Three Dimension
Additive Schwarz Methods for Elliptic Finite Element Problems in Three Dimension
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1. For all u G V'''(r), with To and T, j defined by (4. 9), we have jj (4. 10) 7o(l + \og—)-K^{u, u) < s{Tu, u) < j, s{u, u) Here 70 and 71 are constants independent of H and h and the jumps of the coefficients. To prove this theorem, we use Theorem 3. 1 of Section 3; cf. Dryja, Smith and Widlund [4]. For an alternative proof, see Smith [9], [10]. We now briefly discuss how the method can be implemented. Problem (2. 14) has been replaced by (4. 11) Tu = g where g = go + ES'ij. 9o = TqW/i, and gij = T. /u/i- To solve (4. 11) we can use the conjugate gradient method since T is symmetric, positive definite and well conditioned. For simplicity we only consider the first Richardson method. Thus ^ ^, "+1 = i, n _ ^rp^^n _ y) ^ ^ ^ 2/(70(1 + log -r)-' + 7l) n Let r" = r(u" - tz) = rS + Eij rf^, where r^ = ro(u" - u) and rf^ = r. J(u" - u). To find Tq and r^j, we solve (4. 12a) boir^, v) = ^(u", u) -{g, v) = F{v), v e Vo and (4. 126) b, j{r^^, v) = Fiv), vGV„ To compute s{u"-, v) we solve the Dirichlet problems (4.
What to read after Additive Schwarz Methods for Elliptic Finite Element Problems in Three Dimension?
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To quickly assess the difficulty of the text, read a short excerpt:
1. For all u G V'''(r), with To and T, j defined by (4. 9), we have jj (4. 10) 7o(l + \og—)-K^{u, u) < s{Tu, u) < j, s{u, u) Here 70 and 71 are constants independent of H and h and the jumps of the coefficients. To prove this theorem, we use Theorem 3. 1 of Section 3; cf. Dryja, Smith and Widlund [4]. For an alternative proof, see Smith [9], [10]. We now briefly discuss how the method can be implemented. Problem (2. 14) has been replaced by (4. 11) Tu = g where g = go + ES'ij. 9o = TqW/i, and gij = T. /u/i- To solve (4. 11) we can use the conjugate gradient method since T is symmetric, positive definite and well conditioned. For simplicity we only consider the first Richardson method. Thus ^ ^, "+1 = i, n _ ^rp^^n _ y) ^ ^ ^ 2/(70(1 + log -r)-' + 7l) n Let r" = r(u" - tz) = rS + Eij rf^, where r^ = ro(u" - u) and rf^ = r. J(u" - u). To find Tq and r^j, we solve (4. 12a) boir^, v) = ^(u", u) -{g, v) = F{v), v e Vo and (4. 126) b, j{r^^, v) = Fiv), vGV„ To compute s{u"-, v) we solve the Dirichlet problems (4.
What to read after Additive Schwarz Methods for Elliptic Finite Element Problems in Three Dimension?
You can find similar books in the "Read Also" column, or choose other free books by Maksymilian Dryja to read online
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