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 [45]:

* 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*.