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

We derive optimal order a posteriori error estimates in the $L^{\mathrm{\infty }}(L^2)$ and $L^1(L^2)$-norms for the fully discrete approximations of time fractional parabolic differential equations. For the discretization in time, we use the $L1$ methods, while for the spatial discretization, we use standard conforming finite element methods. The linear and quadratic space-time reconstructions are introduced, which are generalizations of the elliptic space reconstruction. Then the related a posteriori error estimates for the linear and quadratic space-time reconstructions play key roles in deriving global and pointwise final error estimates. Numerical experiments verify and complement our theoretical results.