Preconditioning strategies for elliptic partial differential operators and corresponding difference operators are compared by way of a model problem. The technique considered here makes use of a selfadjoint positive definite operator B as the preconditioner for the nonselfadjoint convection -diffusion operator A. Preconditioning by A’s leading term plus a positive zeroth-order term $\sigma I$ and optimizing over $\sigma $ is proposed. The analysis of a model problem indicates that when A has large first-order terms, both the spectral and norm condition of $B^{ - 1} A$ are minimized by $\sigma \gg 0$.