Least-squares spectral element methods are based on two important and successful numerical methods: spectral/hp element methods and least-squares finite element methods. In this respect, least-squares spectral element methods seem very powerful since they combine the generality of finite element methods with the accuracy of the spectral methods and also have the theoretical and computational advantages of the least-squares methods. These features make the proposed method a competitive candidate for the solution of large-scale problems arising in scientific computing. In order to demonstrate its competitiveness, the method has been applied to an analytical problem and the theoretical optimal and suboptimal a priori estimates have been confirmed for various boundary conditions. Moreover, the exponential convergence rates, typical for a spectral element discretization, have also been confirmed. The comparison with the classical Galerkin spectral element method revealed that the least-squares spectral element method is as accurate as the Galerkin method for the smooth model problem.