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Building on the successes of local kernel methods for approximating thesolutions to partial differential equations (PDE) and the evaluation ofdefinite integrals (quadrature/cubature), a local estimate of the error in suchapproximations is developed. This estimate is useful for determining locationsin the solution domain where increased node density (equivalently, reduction inthe spacing between nodes) can decrease the error in the solution. An adaptiveprocedure for adding nodes to the domain for both the approximation ofderivatives and the approximate evaluation of definite integrals is described.This method efficiently computes the error estimate at a set of prescribedpoints and adds new nodes for approximation where the error is too large.Computational experiments demonstrate close agreement between the errorestimate and actual absolute error in the approximation. Such methods arenecessary or desirable when approximating solutions to PDE (or in the case ofquadrature/cubature), where the initial data and subsequent solution (orintegrand) exhibit localized features that require significant refinement toresolve and where uniform increases in the density of nodes across the entirecomputational domain is not possible or too burdensome.

Adaptivity in Local Kernel Based Methods for Approximating the Action of Linear Operators