In this talk we consider two related approaches for solving linear systems that arise from a higher-order finite element discretization of elliptic partial differential equations. The first approach explores direct application of an algebraic-based multigrid method (AMG) to iteratively solve the linear systems that result from higher-order discretizations. While the choice of basis used on the discretization has a significant impact on the performance of the solver, results indicate that AMG is capable of solving operators from both Poisson's equation and a first-order system least-squares (FOSLS) formulation of Stoke's equation in a scalable manner, nearly independent of basis order, p. The second approach incorporates preconditioning based on a bilinear finite element mesh overlaying the entire set of degrees of freedom in the higher-order scheme. AMG is applied to the operator based on the bilinear finite element discretization and is used as a preconditioner in a conjugate gradient (CG) iteration to solve the algebraic system derived from the high-order discretization. This approach is also nearly independent of p. We present several numerical examples that support each method and discuss the computational advantages of the preconditioning implementation.