The finite-element method is applied to field problems governed by Laplace’s equation and, in particular, to potential flow in fluid mechanics. The conditions under which the variational method may be used are examined for Dirichlet, Neumann, and mixed boundary conditions, and for both singly and multiply connected regions. The discretization of the field, using finite elements of triangular form is developed, and the resulting equations are solved. A computer program based on this analysis has been developed, and will solve any two-dimensional potential fields for simple or mixed boundary conditions and for singly or multiply connected regions. It may be used for multiple-body flow fields, such as aerofoil cascades, with boundary constraints such as the Kutta condition.