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 [4] and [11] for some techniques of the anisotropic

interpolation error estimates.

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 estimate

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.