Preconditioning and Boundary Conditions

Abstract

Consider the large systems of linear equations $A_h u_h = f_h $ that arise from the discretization of a second-order elliptic boundary-value problem. Consider also the preconditioned systems (i) $B_h^{ - 1} A_h u_h = B_h^{ - 1} f_h $ and (ii) $A_h B_h^{ - 1} v_h = f_h $, $u_h = B_h^{ - 1} v_h $, where $B_h $ is itself a matrix that arises from the discretization of another elliptic operator. The effect of boundary conditions (of A and B) on the $L_2 $ and $H_1 $ condition of $B_h^{ - 1} A_h $, $A_h B_h^{ - 1} $ is discussed. In particular, in the case of $H_2 $ regularity, it is found that $\| {B_h^{ - 1} A_h } \|_{L_2 } $ is uniformly bounded if and only if $A^ * $ and $B^ * $ have the same boundary conditions, whereas $\| {A_h B_h^{ - 1} } \|_{L_2 } $ is uniformly bounded if and only if A and B have the same boundary conditions. Similarly, $\| {B_h^{ - 1} A_h } \|_{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 $H_2 $ regularity.