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A linearized numerical scheme is proposed to solve the nonlinear timefractional parabolic problems with time delay. The scheme is based on thestandard Galerkin finite element method in the spatial direction, thefractional Crank-Nicolson method and extrapolation methods in the temporaldirection. A novel discrete fractional Gr\"{o}nwall inequality is established.Thanks to the inequality, the error estimate of fully discrete scheme isobtained. Several numerical examples are provided to verify the effectivenessof the fully discrete numerical method.

Fractional Crank-Nicolson-Galerkin finite element methods for nonlinear time fractional parabolic problems with time delay