Error Analysis of a Compact ADI Scheme for the 2D Fractional Subdiffusion Equation

In this paper, a Crank–Nicolson-type compact ADI scheme is proposed for solving two-dimensional fractional subdiffusion equation. The unique solvability, unconditional stability and convergence of the scheme are proved rigorously. Two error estimates are presented. One is $$\mathcal{O }(\tau ^{\min \{2-\frac{\gamma }{2},\,2\gamma \}}+h_1^4+h^4_2)$$O(τmin{2-γ2,2γ}+h14+h24) in standard $$H^1$$H1 norm, where $$\tau $$τ is the temporal grid size and $$h_1,h_2$$h1,h2 are spatial grid sizes; the other is $$\mathcal{O }(\tau ^{2\gamma }+h_1^4+h^4_2)$$O(τ2γ+h14+h24) in $$H^1_{\gamma }$$Hγ1 norm, a generalized norm which is associated with the Riemann–Liouville fractional integral operator. Numerical results are presented to support the theoretical analysis.

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