  This paper studies adaptive first-order least-squares finite element methods
for second-order elliptic partial differential equations in non-divergence
form. Unlike the classical finite element method which uses weak formulations
of PDEs not applicable for the non-divergence equation, the first-order
least-squares formulations naturally have stable weak forms without using
integration by parts, allow simple finite element approximation spaces, and
have build-in a posteriori error estimators for adaptive mesh refinements.
  The non-divergence equation is first written as a system of first-order
equations by introducing the gradient as a new variable. Then two versions of
least-squares finite element methods using simple $C^0$ finite elements are
developed in the paper, one is the $L^2$-LSFEM which uses linear elements, the
other is the weighted-LSFEM with a mesh-dependent weight to ensure the optimal
convergence. Under a very mild assumption that the PDE has a unique solution,
optimal a priori and a posteriori error estimates are proved. With an extra
assumption on the operator regularity which is weaker than traditionally
assumed, convergences in standard norms for the weighted-LSFEM are also
discussed. $L^2$-error estimates are derived for both formulations. We perform
extensive numerical experiments for smooth, non-smooth, and even degenerate
coefficients on smooth and singular solutions to test the accuracy and
efficiency of the proposed methods.
