Diagonal Scalings of the Laplacian as Preconditioners for Other Elliptic Differential Operators

Abstract

The use of diagonal scalings of the Laplacian matrix as preconditioners for matrices arising from other second-order self adjoint elliptic differential operators is considered. It is proved that if a diffusion operator with a piecewise constant but discontinuous diffusion coefficient is preconditioned by a diagonal scaling of the Laplacian, then, in the limit as the mesh size goes to zero, the optimal diagonal scaling is just the identity. If, on the other hand, the Laplacian is scaled on each side by the square root of the diagonal of the matrix corresponding to the diffusion operator, then the condition number of the preconditioned matrix grows like $O(h^{ - 2} )$, instead of $O(1)$. This is in contrast to the case in which the diffusion coefficient is smoothly varying, in which case numerical evidence suggests that the optimal diagonal scaling is approximately equal to the square root of the diagonal of the matrix.