PRECONDITIONING AND BOUNDARY-CONDITIONS

Consider the large systems of linear equations Ahuh = fh that arise from the discretization of a second-order elliptic boundary-value problem. Consider also the preconditioned systems (i) Bh-1 Ahuh = Bh-1fh and (ii) AhBh-1vh = fh, uh = Bh-1vh, where Bh is itself a matrix that arises from the discretization of another elliptic operator. The effect of boundary conditions (of A and B) on the L2 and H1 condition of Bh-1Ah,AhBh-1 is discussed. In particular, in the case of H2 regularity, it is found that ∥Bh-1Ah∥L(2) is uniformly bounded if and only if A and B have the same boundary conditions, whereas ∥AhBh-1∥L(2) is uniformly bounded if and only if A and B have the same boundary conditions. Similarly, ∥Bh-1Ah∥H(1) is uniformly bounded if and only if A and B have homogeneous Dirichlet boundary conditions on the same portion of the boundary. This latter result does not depend on H2 regularity.