Abstract: | We develop a new least squares method for solving the second-order elliptic equations in non-divergence form. Two least-squares-type functionals are proposed for solving the equation in two sequential steps. We first obtain a numericalapproximation to the gradient in a piecewise irrotational polynomial space. Thentogether with the numerical gradient, we seek a numerical solution of the primitivevariable in the continuous Lagrange finite element space. The variational settingnaturally provides an a posteriori error which can be used in an adaptive refinementalgorithm. The error estimates under the $L^2$ norm and the energy norm for bothtwo unknowns are derived. By a series of numerical experiments, we verify theconvergence rates and show the efficiency of the adaptive algorithm. |