  This paper deals with the numerical computation of null controls for the
linear heat equation. The goal is to compute approximations of controls that
drive the solution from a prescribed initial state to zero at a given positive
time. In [Fernandez-Cara \& M\"unch, Strong convergence approximations of null
controls for the 1D heat equation, 2013], a so-called primal method is
described leading to a strongly convergent approximation of distributed
control: the controls minimize quadratic weighted functionals involving both
the control and the state and are obtained by solving the corresponding
optimality conditions. In this work, we adapt the method to approximate the
control of minimal square integrable-weighted norm. The optimality conditions
of the problem are reformulated as a mixed formulation involving both the state
and its adjoint. We prove the well-posedness of the mixed formulation (in
particular the inf-sup condition) then discuss several numerical experiments.
The approach covers both the boundary and the inner situation and is valid in
any dimension.
