Preconditioning and Boundary Conditions without $H_2$ Estimates: $L_2$ Condition Numbers and the Distribution of the Singular Values
| Autor: | Seymour V. Parter, Thomas A. Manteuffel, Charles I. Goldstein |
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| Rok vydání: | 1993 |
| Předmět: | |
| Zdroj: | SIAM Journal on Numerical Analysis. 30:343-376 |
| ISSN: | 1095-7170 0036-1429 |
| DOI: | 10.1137/0730017 |
| Popis: | This work deals with the behavior - in the L[sub 2] norm - of the condition number and distribution of the L[sub 2] singular values of the preconditioned operators B[sub h][sup [minus]1]A[sub h] and A[sub h]B[sub h][sup [minus]1][sub h], where A[sub h] and B[sub h] are finite element discretizations of second-order elliptic operators, A and B. In an earlier work, Manteuffel and Parter proved that B[sub h][sup [minus]1]A[sub h] (A[sub h]B[sub h][sup [minus]1]) have a uniformly bounded L[sub 2] condition number if and only if A[sup *] and B[sup *] (A and B) have the same boundary conditions. This earlier work used the H[sub 2] regularity of A and B, as well as optimal L[sub 2] error estimates and a quasi-uniform grid for the finite element spaces. In the present paper, the authors first extend these condition number results to the case in which neither H[sub 2] regularity (and hence optimal L[sub 2] error estimates) nor the quasi-uniformity assumption need to be satisfied. Instead, it is assumed that the principle part of the preconditioning operator B is a scalar multiple, 1/[mu], of the principle part of A. |
| Databáze: | OpenAIRE |
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