Preconditioning and boundary conditions without H 2 estimates: L 2 condition numbers and the distribution of the singular values

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.