  We show that the mass matrix derived from finite elements can be effectively
used as a preconditioner for iteratively solving the linear system arising from
finite-difference discretization of the Poisson equation, using the conjugate
gradient method. We derive analytically the condition number of the
preconditioned operator. Theoretical analysis shows that the ratio of the
condition number of the Laplacian to the preconditioned operator is $8/3$ in
one dimension, $9/2$ in two dimensions, and $2^9/3^4 \approx 6.3$ in three
dimensions. From this it follows that the expected iteration count for
achieving a fixed reduction of the norm of the residual is smaller than a half
of the number of the iterations of unpreconditioned CG in 2D and 3D. The scheme
is easy to implement, and numerical experiments show its efficiency.
