This paper proves that the approximation of pointwise derivatives of order s of functions in Sobolev space W-2(m) (R-d) by linear combinations of function values cannot have a convergence rate better than m - s - d/2,...
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This paper proves that the approximation of pointwise derivatives of order s of functions in Sobolev space W-2(m) (R-d) by linear combinations of function values cannot have a convergence rate better than m - s - d/2, no matter how many nodes are used for approximation and where they are placed. These convergence rates are attained by scalable approximations that are exact on polynomials of order at least [m - d/2] + 1, proving that the rates are optimal for given m, s and d. And, for a fixed node set X subset of R-d, the convergence rate in any Sobolev space W-2(m) (Omega) cannot be better than q - s where q is the maximal possible order of polynomial exactness of approximations based on X, no matter how large m is. In particular, scalable stencil constructions via polyharmonic kernels are shown to realize the optimal convergence rates, and good approximations of their error in Sobolev space can be calculated via their error in Beppo-Levi spaces. This allows us to construct near-optimal stencils in Sobolev spaces stably and efficiently, for use in meshless methods to solve partial differential equations via generalized finite differences. Numerical examples are included for illustration.
Let B be a convex polytope in the d-dimensional Euclidean space. We consider an interpolation of a function f at the vertices of B and compare it with the interpolation of f and its derivative at a fixed point y is an...
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Let B be a convex polytope in the d-dimensional Euclidean space. We consider an interpolation of a function f at the vertices of B and compare it with the interpolation of f and its derivative at a fixed point y is an element of B. The two methods may be seen as multivariate analogues of an interpolation by secants and tangents, respectively. For twice continuously differentiable functions, we establish sharp error estimates with respect to a generalized L-p norm for 1 <= p <= infinity. The case p = 1 is of special interest since it provides analogues of the midpoint rule and the trapezoidal rule for approximate integration over the polytope P. In the case where P is a simplex and p > 1, this investigation covers recent results by S. Waldron [SIAM J. Numer. Anal., 35 (1998), pp. 1191-1200] and by M. Stampfle [J. Approx. Theory, 103 (2000), pp. 78-90].
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