The optimal amount of diffusion is found in 4 steps. First the monotonicityproblem is recast in the form of a maximum principle. Secondly, for a 2 x 2element matrix, the amount of diffusion is calculated for an arbitraryright-hand side so that the solution obeys a maximum principle. Thirdly, theresult is generalised for larger matrices. And finally, the result is recastto meet the monotonicity requirement. The result is an equation giving theamount of diffusion to be added in terms of a given right-hand-side vector.Intuitively, this diffusion is related tothe local "curvature" of the right-hand side.Selective lumping is shown to be effective for both an Eulerian-Lagrangianlocalized adjoint method (ELLAM) solution of the transport equation and afinite element solution of the heat equation. In both cases, solutions aremonotonic and contain less numerical diffusion than in standard lumpingschemes. The slumping concept is general and can be applied to any numericalapproximation based on the method of weighted residuals. The particularderivation presented here is limited to symmetric tridiagonal Toeplitzmatrices.