Block-centered local refinement methods for the time-fractional equations

Abstract In this paper, we present and analyze two block-centered local refinement (BLR) methods for solving the time-fractional equations. The main difference between the two methods is that different approximation methods are used for the value of the pressure at the slave nodes. One adopts a piecewise constant interpolation approximation (PCIA) method, which is called simple block-centered local refinement (S-BLR) method. The other utilizes a piecewise linear interpolation approximation (PLIA) method, which called more accurate block-centered local refinement (MA-BLR) method. The stability analysis is proved carefully. It is estimated that the discrete L 2 errors for the velocity and pressure are O ( ▵ t 2 − α + h 3 / 2 ) and O ( ▵ t 2 − α + h 3 / 2 ) in use of S-BLR and MA-BLR methods, respectively. Where Δ t is the time step and h is the maximal mesh size. These error estimate results are all established on locally refinement composite grids. Finally, a numerical experiment is presented to show that the convergence rates are in agreement with the theoretical analysis.