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.
