Preconditioning and Boundary Conditions

Consider the large systems of linear equations A h u h = f h which arise from the discretization of a second-order elliptic boundary value problem. Consider also the preconditioned systems (i) Bh − 1 A h u h = Bh 1 f h; and (ii) A h Bh 1 v h = f h , u h = Bh 1 v h where B h is itself a matrix which arises from the discretization of another elliptic operator. We discuss the effect of boundary conditions (of A and B) on the L 2 and H 1 condition of Bh − 1 A h , A h Bh − 1 . In particular, in the case of H 2 regularity one finds that  Bh 1 A h  L 2 is uniformly bounded if and only if A * and B * have the same boundary conditions while  A h Bh 1  L 2 is uniformly bounded if and only if A and B have the same boundary conditions. Similarly,  Bh 1 A h  H 1 is uniformly bounded only if A and B have homogeneous Dirichlet boundary conditions on the same portion of the boundary. This latter result does not depend upon H 2 regularity.