Non-intrusive Least-Squares Functional A Posteriori Error Estimator: Linear and Nonlinear Problems with Plain Convergence

TL;DR

This paper develops a non-intrusive least-squares functional a posteriori error estimator applicable to linear and nonlinear problems, enabling adaptive mesh refinement without requiring the original least-squares method, with proven convergence.

Contribution

It introduces a systematic approach for applying least-squares error estimation to problems not solved by least-squares methods, including a new interpretation and convergence analysis.

Findings

Effective for elliptic PDEs with conforming finite element methods

Extends to nonlinear problems with proven convergence

Simplifies reliability and efficiency analysis of error estimators

Abstract

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.