The classical Monge-Kantorovich problem is to move a given distribution of mass onto another one with the same total in the cheapest way; the transportation cost is assumed equal to distance (L1 problem). We present a mixed numerical scheme based on the known dual variational formulation of this problem in terms of transport flux. Then we consider a generalized (partial) Monge-Kantorovich problem in which only a given amount of mass has to be optimally transported from one distribution onto another and the distributions may be unbalanced. We derive a new variational formulation for the arising free boundary problem in optimal transportation and use it to solve the problem numerically. Both schemes are based on regularization of non-differentiable functionals and Raviart-Thomas element of the least order; the second algorithm employs also an augmented.