Preconditioning and Boundary Conditions

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Abstract
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 - H 1 is uniformly bounded only if A and B have homogeneous,Dirichlet boundary,conditions on the same,portion of the boundary. Thislatter result does not depend upon,H 2 regularity. Key words. preconditioning, elliptic operators, condition number, H2 regularity, boundary,conditions