Finite element approximation to linear, second order, parabolic problems with $L^1$ data

We consider the approximation to the solution of the initial boundary value problem for the heat equation with right hand side and initial condition that merely belong to $L^1$. Due to the low integrability of the data, to guarantee well-posedness, we must understand solutions in the renormalized sense. We prove that, under an inverse CFL condition, the solution of the standard implicit Euler scheme with mass lumping converges, in $L^{\mathrm{\infty }}(0,T;L^1(\mathrm{\Omega }))$ and $L^q(0,T;W_0^{1,q}(\mathrm{\Omega }))$ ($q<\frac{d+2}{d+1}$), to the renormalized solution of the problem.