The regulator equation is the fundamental equation whose solution must be found in order to solve the output regulation problem. It is a system of first-order partial differential equations (PDE) combined with an algebraic equation. The classical approach to its solution is to use the Taylor series with undetermined coe±cients. In this contribution, another path is followed: the equation is solved using the finite-element method which is, nevertheless, suitable to solve PDE part only. This paper presents two methods to handle the algebraic condition: the first one is based on iterative minimization of a cost functional defined as the integral of the square of the algebraic expression to be equal to zero.