Almost optimal order approximate inverse based preconditioners for 3-D convection dominated problems on tensor-grids

For a one-dimensional diffusion problem on an refined computational grid we present preconditioners based on the standard approximate inverse technique. Next, we determine its spectral condition number ¿2 and perform numerical calculations which corroborate the theoretical results. Then we perform numerical calculations which show that the standard approximate inverse preconditioners and our modified versions behave in a similar manner. To finish with we show that a combination of the standard approximate inverse with an additional incomplete factorisation leads to an almost optimal order preconditioner in 1–3 dimensions on refined grids, with and without dominant convection.