Before revision:  We approximately factorize the diagonal blocks in these preconditioners using a near-linear-complexity multilevel ILUfactorization called HILUCSI, which is significantly more robust and more efficient than ILU(0)  . We apply the  preconditioners in right-preconditioned GMRES to solve the advection-diffusion equation in 3D using finite element and finite difference methods.
After revision:  We approximately factorize the diagonal blocks in these preconditioners using a near-linear-complexity multilevel ILUfactorization called HILUCSI, which is significantly more robust and more efficient than ILU(0) . We apply the block preconditioners in right-preconditioned GMRES to solve the advection-diffusion equation in 3D using finite element and finite difference methods.