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 κ2 and perform numerical calculations which corroborate the result. 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 factorization leads to an almost optimal order preconditioner in 1, 2 and 3 dimensions, with or without dominant convection.