The exact order of convergence for finite difference approximations to ordinary boundary value problems

Abstract:

This paper deals with the problem of determining the exact order of convergence for the finite difference method applied to ordinary boundary value problems when formulas of different orders are used at different points of the grid. Under rather general assumptions, it is shown that the global discretization error is $O({h^\tau })$ if the local truncation error is $O({h^\tau })$ on the boundary and at interior grid points, while it is only $O({h^{\tau - (k - \mu )}})$ at grid points near the boundary. Here k and $\mu$ denote the order of the differential and the boundary operator, respectively.