The FINESSE code (finite element solver for stationary equilibria) computes axisymmetric magnetohydrodynamic equilibria in poloidal elliptic flow regimes for a variety of astrophysical and laboratory plasma configurations. The obtained equilibria are accurate and are used to study the spectral characteristics of such flowing equilibria. The nonlinear partial differential equation for the poloidal magnetic flux is solved in a weak form via Picard iteration, resulting in a large-scale linear problem. The algebraic Bernoulli equation for the poloidal Alfvén Mach number is solved with a nonlinear root finder. Converged solutions are obtained by iterating on these two equations.