A technical description of the commutation problem.

Let T be a triangulation and S(T) a space of smooth piecewise polynomial functions defined on T . We denote by r the degree of smoothness of the functions in S and by d their polynomial degree. Of particlar interest is the case

r=e and d=3. (1)

We are interested in those functions in S that interpolate to given values f(V) at the vertices V of T, and in particular wish to find that interpolating function c s thet minimizes a variational principle I, i.e.,

I(s)= min subject to s(V)=f(V) for all V in T (2)

The most frequent choice of I is the clamped elastic plate functional, i.e.,

(3)

The dimension of S often depends on the precise location of the vertices of T. The solution of (1) also depends on T. Our key question is: Does s depend on T continuously?

Consider the following diagram

(4)

T0 denotes the triangulation where the dimesnion of S increases. s(T) denotes the solution of (2). One can now go from s(T) to s(T0) in two different ways:

1. Consider the limit of s(T) as T tends to T0, or
2. Let T=T0 and solve (1).

In terms of the diagram (4), we go from the Northwest corner to the Southeast corner, first vetrtically in the first approach, and first horizontally in the second approach.

One can now ask the following questions:

1. Does s(T) have a limit as T approaches T0?
2. If it does, does the limit equal the solution of (2) for the case T=T0? In other words,
3. Does the diagram (4) commute?

The last question gives the name to the whole problem.