Example 1

This is the standard clamped elastic plate functional and the space S of C1 cubics. We consider first the cardinal function associated with a boundary vertex (V1). Here

is a view of the interpolant for the interior vertex being at singularity. Color indicates the function value. For all pictures, just click to see an enlarged version.

Sequence 1

How do we know that in this case the diagram commutes? Strictly speaking we don't, but we have numerical evidence. Consider a sequence of images where the interior vertex moves towards singularity, and compute the difference between the interpolant and the interpolant at singularity. The maximum difference approaches zero as the vertex approaches singularity. These observations can be confirmed and refined by computing and displaying the maxim deviations.

This is illustrated in the following sequence, where color indicates the deviation from singularity, within an interval from -0.001 to 0.001. The location of the interior vertex and the minimum and maximum values of the deviation are also given. The following sequences of pictures is obtained:

1. V5=(1/64,0), discrepancy = [-0.005,0.000007]. The red color over most of the surface the indicates that the surface at singularity is more than 0.001 above the surface prior to singularity.
2. V5=(1/128,0), discrepancy = [-0.002,0.000002]. The red areas has shrunk, ie.., the surface is getting closer to the surface at singularity.
3. V5=(1/256,0) discrepancy = [-0.001,0.0000005]. The process continues
4. V5=(1/512,0), discrepancy = [-0.0005,0.0000001] ... and continues.
5. V5=(1/1024,0), discrepancy = [-0.0003,0.00000003] . At this stage the surface is withing 0.001 of the singular surface everywhere.
6. V5=(1/2048,0) discrepancy = [-0.0001,0.00000001]. The fit is even better.

Sequence 2

We obtain a similar sequence for the cardinal function associated with the interior vertex:

1. V5=(1/2,1/4), discrepancy = [-0.19,0.34]. This time the color indicates that the surface lies both above and below the singular surface. The vertex apporaches singularity from the direction eta = xi/2. (Letting eta=0 as in the preceding sequence would generate a surface that's constant in xi and that exhibits a basically one-dimensional phenomenon. However, the qualitative behavior would be the same.) The small spots exhibiting colors other than red or violet indicate regions where the two surfaces intersect and are very close. Watch them grow as the interior vertex approaches singularity.
2. V5=(1/4,1/8), discrepancy = [-0.10,0.15].
3. V5=(1/8,1/16), discrepancy = [-0.06,0.07].
4. V5=(1/16,1/32), discrepancy = [-0.03,0.03].
5. V5=(1/32,1/64), discrepancy = [-0.02,0.02].
6. V5=(1/64,1/128), discrepancy = [-0.008,0.008].
7. V5=(1/128,1/256), discrepancy = [-0.004,0.004].
8. V5=(1/256,1/512), discrepancy = [-0.002,0.002]. The close regions are now beginning to dominate the picture.
9. V5=(1/512,1/1024), discrepancy = [-0.001,0.001 ].
10. V5=(1/1024,1/2048), discrepancy = [-0.0005,0.0005].
11. V5=(1/2048,1/4096), discrepancy = [-0.0002,0.0002].

Sequence 3

There is another way to view these changing durfaces, by letting the coloring cover the range from minimum to maximum discrepancy. In that case the colors don't change much until the surface bexomes (numerically singular) and the discrepancy consists of small rando, numbers. To recognize the commutation it is then necessary to pay close attention to the legend in the picture which gives the numerical range of the discrepancy.

Following is a sequence for the interpolant to the data

f(V1)=f(V5)=1, f(V2)=f(V3)=f(V4) =0.

First, to orient ourselves, here are a sequence of pictures (at singularity) colored by

1. --- f
2. --- x
3. --- y
4. --- and index number of triangle.

Now let's look at some pictures colored by discrepancy, and showing the change as the interior vertex approaches singularity.

1. V5 = (1/2,0). The deviation is quite high, up to 20 percent or so. Moving V5 closer to singularity is going to change the shape of the surface and its coloration.
2. V5 = (1/4,0). Interestingly, halving the distance of V5 from the singular vertex also halves the maximum of the discrepancy.
3. V5 = (1/8,0).
4. V5 = (1/16,0).
5. V5 = (1/32,0).
6. V5 = (1/64,0).
7. V5 = (1/4096,0).
8. V5 = (1/8192,0).
9. V5 = (1/16384,0).
10. V5 = (1/32768,0).