The authors consider an open bounded set \(\Omega \subset \mathbb{R}^2\) with smooth enough boundary \(\partial \Omega\) in which \(\partial_u \) is a relatively open portion. Let \(H_{\partial u}^1(\Omega)\) be the set of \(H^1(\Omega)\)-functions that vanish on \(\partial u\). Furthermore, let \(f\) be in the dual of \(H_{\partial u}^1(\Omega)\). The mixed Poisson problem in variational form then consists in minimizing \(\int_{\Omega} | \nabla u| ^2 dx -2 \) over \(u \in H_{\partial u}^1(\Omega)\). This is, in a sense, the simplest problem in finite element theory. The authors approximate the solution by approximating the problem by a sequence of unconstrained minimum problems. So no additional variables and constraint equations or penalizing terms are introduced. The convergence results are proved in a topological sense. The authors also give several numerical examples that illustrate the convergence and allow to estimate the convergence rate.