For the convergence of the above norms to be valid, we need to make sure our basis functions meet certain criteria, namely those of completeness and continuity. These can be stated as follows :
Completeness condition. The element trial solution, , and any of its derivatives up to order m appearing in the integrals of the Galerkin (or variational/weak) formulation, should be able to assume any constant value within an element when, in the limit, the size of the element decreases to zero.
Continuity condition. At interelement boundaries, the element trial solutions should be -continuous; that is, and its derivatives up to order m-1 should be continuous.
If the element basis functions, , are constructed so that satifies both conditions, then a sequence of approximate solutions, corresponding to a sequence of successively refined meshes, will converge energywise to the exact solution (again, assuming no other errors) in the limit as , where h represents the size of the element. This is often referred to as h-convergence.