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We study a fully discrete finite element method for variable-ordertime-fractional diffusion equations with a time-dependent variable order.Optimal convergence estimates are proved with the first-order accuracy in time(and second order accuracy in space) under the uniform or graded temporal meshwithout full regularity assumptions of the solutions. Numerical experiments arepresented to substantiate the analysis.

Optimal order finite element approximations for variable-order time-fractional diffusion equations