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.

翻译：基于局部核方法在逼近偏微分方程解和定积分求积（数值积分）计算中的成功应用，本文发展了这类逼近中误差的局部估计方法。该误差估计对于确定解域中需要增加节点密度（即减小节点间距）以降低解误差的位置具有实用价值。文中描述了一种自适应程序，用于在导数的逼近和定积分的近似计算中向解域添加节点。该方法能高效计算预设点集处的误差估计，并在误差过大处新增逼近节点。计算实验表明，误差估计与逼近中的实际绝对误差高度吻合。当逼近偏微分方程解（或处理求积/数值积分情形）时，若初始数据及后续解（或被积函数）呈现须通过显著细化网格才能解析的局部特征，且在整个计算域内均匀增加节点密度不可行或过于繁重时，此类方法具有必要性或可取性。