Numerical solution of a free boundary problem associated to investments with instantaneous irreversible environmental effects

In this work, some numerical methods are proposed to solve a problem which models the joint utility of starting at optimal time an industrial process that provides some benefits but also implies some irreversible effects on the environment. The mathematical model is posed in terms of an obstacle problem associated to an elliptic equation on an unbounded domain. First, following Díaz and Faghloumi [5], an equivalent formulation on a bounded domain is proposed by means of adequate changes of variables and unknown. Next, projected Gauss–Seidel and Lions–Mercier type algorithms to solve the finite elements discretized problem are proposed; the latter is combined with multigrid techniques and adaptive refinement. The algorithms efficiency is first illustrated by some numerical tests on examples with known analytical solution or theoretically stated properties. Finally, some more realistic examples are presented for which there are not known properties of the solution.