  In this paper, the time fractional reaction-diffusion equations with the
Caputo fractional derivative are solved by using the classical $L1$-formula and
the finite volume element (FVE) methods on triangular grids. The existence and
uniqueness for the fully discrete FVE scheme are given. The stability result
and optimal \textit{a priori} error estimate in $L^2(\Omega)$-norm are derived,
but it is difficult to obtain the corresponding results in $H^1(\Omega)$-norm,
so another analysis technique is introduced and used to achieve our goal.
Finally, two numerical examples in different spatial dimensions are given to
verify the feasibility and effectiveness.
