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• W2001837653 abstract "This work deals with the behavior—in the $L_2$ norm—of the condition number and distribution of the $L_2$ singular values of the preconditioned operators $B_h^{ - 1} A_h $ and $A_h B_h^{ - 1} $, where $A_h$ and $B_h$ are finite element discretizations of second-order elliptic operators, A and B. In an earlier work, Manteuflel and Parter [SIAM J. Numer. Anal., 27 (1989), pp. 656–694] proved that $B_h^{ - 1} A_h (A_h B_h^{ - 1} )$ have a uniformly bounded $L_2$ condition number if and only if $A^ * $ and $B^ * $ (A and B) have the same boundary conditions. This earlier work used the $H_2$ regularity of A and B, as well as optimal $L_2$ error estimates and a quasi-uniform grid for the finite element spaces. In the present paper, we first extend these condition number results to the case in which neither $H_2$ regularity (and hence optimal $L_2$ error estimates) nor the quasi-uniformity assumption need be satisfied. Instead, it is assumed that the principal part of the preconditioning operator B is a scalar multiple, ${1 / mu }$, of the principal part of A. It is also proven in this case that the operators $Q = B^{ - 1} A - mu I$ and $tilde Q = AB^{ - 1} mu I$ are compact, and the corresponding discrete operators are collectively compact and consistent approximations to Q and $tilde Q$. Using this, it is shown that the $L_2$ singular values of $B_h^{ - 1} A_h $ and $A_h B_h^{ - 1} $ “fill” the interval $[mu _0 ,mu _1 ]$ where $mu _0 > 0$ and $mu _1 $ are the minimum and maximum values of $mu $. Moreover, for any $varepsilon > 0$ there are (at most) a finite number, $n(varepsilon )$, of singular values outside the interval $[mu _0 - varepsilon ,mu _1 + varepsilon ]$. Analogous results are also proven for the case when $B_h^{ - 1} $ is replaced by a more practical preconditioner, say $dot B_h^{ - 1} $, which is equivalent to $B_h^{ - 1} $ in the $L_2$ norm. This has important implications for the solution of the preconditioned discrete equations using the conjugate gradient method. In particular, the convergence rate will be better than the usual bound obtained using condition number estimates. Finally, some matrix implementations for both the left and right preconditioned normal equations are discussed in detail. These include implementations that avoid the inversion of the mass matrix." @default.
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• W2001837653 date "1993-04-01" @default.
• W2001837653 modified "2023-10-14" @default.
• W2001837653 title "Preconditioning and Boundary Conditions without $H_2$ Estimates: $L_2$ Condition Numbers and the Distribution of the Singular Values" @default.
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• W2001837653 doi "https://doi.org/10.1137/0730017" @default.
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