Non-intrusive Least-Squares Functional A Posteriori Error Estimator: Linear and Nonlinear Problems with Plain Convergence
Ziyan Li, Shun Zhang
TL;DR
This work extends the least-squares functional error estimator to non-intrusive use with problems not solved by LSFEM, by formulating a solve-recover (two-step) framework and analyzing it within an abstract AFEM setting. It develops a non-intrusive estimator by recovering an auxiliary flux via LS minimization with the primal solution fixed, and proves reliability, efficiency, and plain convergence for both linear second-order elliptic equations and a simple monotone nonlinear model. The approach yields an estimator that can drive adaptive mesh refinement even when the underlying discretization is not based on a least-squares formulation, with rigorous a priori and a posteriori analyses for the combined solve-recover problem. The results provide a practical, broadly applicable tool for error control and mesh adaptivity in linear and nonlinear elliptic problems, with prospects for extension to more complex systems such as Navier–Stokes.
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.