Our procedure for formulating the individual element equations from a variational principle and our privilege to assemble these equations to obtain the system's (global) equations rely on the assumption that the interpolation functions satisfy certain requirements. The requirements we place on the choice of interpolation functions stem from the need to ensure that our approximate solution converges to the correct one when we use an increasing number of smaller elements, that is, when we refine the element mesh. Mathematical proofs of convergence assume that the process of mesh refinement occurs in a regular fashion as follows:

1. • The elements must be made smaller in such a way that every point of the solution domain can always be within an element regardless of how small the element might be.

2. • All previous meshes must be contained in the refined meshes.

3. • The form of interpolation functions must remain unchanged during mesh refinement.

These three conditions are illustrated in Figure 4.1, where a simple two dimensional solution domain in the form of an equilateral triangle is discretized with an increasing number of three-noded triangles. We note that when elements with straight boundaries are used to model solution domains with curved boundaries, the first two conditions are not satisfied, and rigorous mathematical proofs of convergence may not be obtainable. Despite this limitation, many applications of the finite element method to problems with non-polygonal solution domains yield acceptable engineering solutions.