A backward Euler alternating direction implicit difference scheme for the three-dimensional fractional evolution equation

A backward Euler alternating direction implicit (ADI) difference scheme is formulated and analyzed for the three-dimensional fractional evolution equation. In our method, the Riemann-Liouville fractional integral term is treated by means of first order convolution quadrature suggested by Lubich. Meanwhile, an ADI technique is adopted to reduce the multidimensional problem to a series of one-dimensional problems. A fully discrete difference scheme is constructed with space discretization by finite difference method. Two new inner products and corresponding norms are defined to analyze the scheme. The verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature. Numerical experiments are reported to demonstrate the efficiency of our scheme.