The a posteriori error estimator using the least-squares functional can be used for adaptive mesh refinement and error control even if the numerical approximations are not obtained from the corresponding least-squares method. This suggests the development of a versatile non-intrusive a posteriori error estimator. In this paper, we present a systematic approach for applying the least-squares functional error estimator to linear and nonlinear problems that are not solved by the least-squares finite element methods. For the case of an elliptic PDE solved by the standard conforming finite element method, we minimize the least-squares functional with conforming approximation inserted to recover the other physical meaningful variable. By combining the numerical approximation from the original method with the auxiliary recovery approximation, we construct the least-squares functional a posteriori error estimator. Furthermore, we introduce a new interpretation that views the non-intrusive least-squares functional error estimator as an estimator for the combined solve-recover process. This simplifies the reliability and efficiency analysis. We extend the idea to a model nonlinear problem. Plain convergence results are proved for adaptive algorithms of the general second order elliptic equation and a model nonlinear problem with the non-intrusive least-squares functional a posteriori error estimators.

翻译：利用最小二乘泛函的后验误差估计器可用于自适应网格细化和误差控制，即使数值近似并非通过相应最小二乘法获得。这为开发通用的非侵入式后验误差估计器提供了思路。本文提出了一种系统方法，将最小二乘泛函误差估计器应用于非最小二乘有限元法求解的线性和非线性问题。针对通过标准协调有限元法求解的椭圆型偏微分方程情形，我们通过插入协调近似来最小化最小二乘泛函，以重构其他具有物理意义的变量。通过结合原始方法的数值近似与辅助重构近似，我们构建了最小二乘泛函后验误差估计器。此外，我们提出了一种新解释，将非侵入式最小二乘泛函误差估计器视为求解-重构组合过程的估计器，从而简化了可靠性与效率分析。我们将该思想推广至模型非线性问题，并证明了采用非侵入式最小二乘泛函后验误差估计器的通用二阶椭圆方程及模型非线性问题自适应算法具有简明收敛性。