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We derive optimal order a posteriori error estimates in the $L^\infty(L^2)$and $L^1(L^2)$-norms for the fully discrete approximations of time fractionalparabolic differential equations. For the discretization in time, we use the$L1$ methods, while for the spatial discretization, we use standard conformingfinite element methods. The linear and quadratic space-time reconstructions areintroduced, which are generalizations of the elliptic space reconstruction.Then the related a posteriori error estimates for the linear and quadraticspace-time reconstructions play key roles in deriving global and pointwisefinal error estimates. Numerical experiments verify and complement ourtheoretical results.

Optimal $L^\infty(L^2)$ and $L^1(L^2)$ a posteriori error estimates for  the fully discrete approximations of time fractional parabolic differential  equations