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

TL;DR

This paper develops an adaptive local kernel method for approximating linear operators, providing error estimates to guide node refinement in solving PDEs and integrals efficiently.

Contribution

It introduces a novel adaptive procedure that estimates errors locally and adds nodes selectively, improving approximation accuracy without uniform refinement.

Findings

Error estimates closely match actual errors in experiments.

Adaptive node addition reduces computational cost compared to uniform refinement.

Method effectively captures localized features in PDE solutions and integrals.

Abstract

Building on the successes of local kernel methods for approximating the solutions to partial differential equations (PDE) and the evaluation of definite integrals (quadrature/cubature), a local estimate of the error in such approximations is developed. This estimate is useful for determining locations in the solution domain where increased node density (equivalently, reduction in the spacing between nodes) can decrease the error in the solution. An adaptive procedure for adding nodes to the domain for both the approximation of derivatives and the approximate evaluation of definite integrals is described. This method efficiently computes the error estimate at a set of prescribed points and adds new nodes for approximation where the error is too large. Computational experiments demonstrate close agreement between the error estimate and actual absolute error in the approximation. Such methods are necessary or desirable when approximating solutions to PDE (or in the case of quadrature/cubature), where the initial data and subsequent solution (or integrand) exhibit localized features that require significant refinement to resolve and where uniform increases in the density of nodes across the entire computational domain is not possible or too burdensome.