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 $L^\infty(0,T;L^1(\Omega))$ and $L^q(0,T;W^{1,q}_0(\Omega))$ ($q<\tfrac{d+2}{d+1}$), to the renormalized solution of the problem.

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