Convergence of $H^1$-Galerkin mixed finite element method for parabolic problems with reduced regularity of initial data

We study the convergence of an $H^1$1-Galerkin mixed finite element method for parabolic problems in one space dimension. Both semi-discrete and fully discrete schemes are analyzed assuming reduced regularity of the initial data. More precisely, for a spatially discrete scheme error estimates of order $\mathcal O(h^2t^{-1/2})$ for positive time are established assuming the initial function $p_0\in H^2(\Omega)\cap H_0^1(\Omega)$. Further, we use an energy technique together with a parabolic duality argument to derive error estimates of order $\mathcal O(h^2t^{-1})$ when $p_0$ is only in $H_0^1(\Omega)$. A discrete-in-time backward Euler method is analyzed and almost optimal order error bounds are established.