On the Conjugate Gradient Solution of the Schur Complement System Obtained from Domain Decomposition

Abstract

For a linear system arising from a finite-element discretization on a quasi-uniform grid of a second-order elliptic partial differential equation, it is shown that the condition number of the Schur complement matrix in domain decomposition is $O({1 / h})$, where h is the mesh parameter. This result is true in both two and three dimensions and regardless of whether the internal separator boundaries have crosspoints. A deflation technique using the restriction to the separator boundaries of the nodal basis for a coarse grid is introduced as a method of preconditioning the Schur complement matrix.