Preconditioning and Boundary Conditions

Consider the large systems of linear equations Aₕ uₕ = fₕ that arise from the discretization of a second-order elliptic boundary-value problem. Consider also the preconditioned systems (i) Bₕ- 1 Aₕ uₕ = Bₕ- 1 fₕ and (ii) Aₕ Bₕ- 1 vₕ = fₕ, uₕ = Bₕ- 1 vₕ, 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- 1 Aₕ, Aₕ Bₕ- 1 is discussed. In particular, in the case of H₂ regularity, it is found that \| Bₕ- 1 Aₕ \|L₂ is uniformly bounded if and only if A^ * and B^ * have the same boundary conditions, whereas \| Aₕ Bₕ- 1 \|L₂ is uniformly bounded if and only if A and B have the same boundary conditions. Similarly, \| Bₕ- 1 Aₕ \|H₁, 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.