where the interpolation error
estimate (2.9), b [member of] [([W.sup.1,[infinity]]([OMEGA])).sup.d], and [c.sub.0] > 0 were used.
We chose to compare the stopping criterion (3.1) with both the exact finite element error and interpolation error
measured in the norms inherited by the problem.
Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is the interpolation error
for the function g(lambda]) = [lambda].sup.j] which vanishes for j = 0, 1, ...
Another argument in favour of the Chebyshev points is given by [12, Theorem 5.6] or [2, Theorem 2.1], which both state that the interpolation error
is of order O([[KAPPA].sup.-n]) for n [right arrow] [infinity], where [KAPPA] is the sum of the semimajor and semiminor axis lengths of f's ellipse of analyticity (in both references the argument is given for the Chebyshev extrema and not the Chebyshev zeros, but asymptotically this does not make any difference).
Observe that the rows tend to stabilize around the underlying Shepard interpolation error
, whereas the columns around the the underlying Xu interpolation error
where for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the centered pth-order interpolation weights from the coarse centers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to [y.sup.(k).sub.j], and [[delta].sup.I] is the interpolation error
, which we bound in [section]4.1 and [section]5.
The interpolation error
estimates on anisotropic triangulations are different to the isotropic case.
On the other hand, most of the former works are concentrated on the estimates of the interpolation error
under anisotropic meshes, in particular, the readers are refer to  and  for some techniques of the anisotropic interpolation error
Now, if f is a compactly supported function in the Sobolev space [W.sup.r+1.sub.[infinity]](R) with Sobolev semi-norm If [[absolute value of f].sup.S.sub.r+1] then the interpolation error
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] satisfies
For the applications in Section 3 we will need interpolation error
estimates in the dual spaces of the Sobolev spaces.
Using (3.5) and (3.7), we can derive the interpolation error
In the case of data loss, especially when the number of data acquisition nodes is relatively small, the interpolation error
of our algorithm is much smaller than that of the inverse-distance-weighted interpolation.