This paper is concerned with conditions for convergence of approximations using the nodal expansion method, a relatively new method for approximate solution of partial differential equations. The essence of the method, which is discussed in the context of a model linear elliptic equation (the Helmholtz equation), is that the domain is partitioned into rectangular elements, and on each element the solution is approximated by a linear combination of Legendre polynomials. A weighted Galerkin method is used to obtain the approximating equations on each element, and on the boundaries of elements. One important result established by the authors is that the nodal expansion method may be formulated as a generalised or nonstandard hybrid variational problem. This hybrid problem is studied in the framework of the Babuška-Brezzi theory for mixed variational problems, and it is shown that finite element solutions are stable and convergent. Solutions to the nonstandard problem are compared with those to the standard hybrid problem corresponding to the elliptic partial differential equation, and in this way it is shown that the nonstandard approximation, and hence the nodal expansion method, yields a solution which converges to the solution of the original problem.