  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.
