The use of high-order spectral and finite elements in the numerical approximation to partial differential equations continues to grow in popularity. In many situations, the accuracy per degree of freedom when using high-order elements surpasses the low-order approach. The total computational cost, however, requires a deeper investigation of the complexity in the full simulation tool chain. In this talk, we focus on a common and nontrivial component of the process: solution of the associated algebraic system of equations. In particular, we highlight an algebraic-based multigrid (AMG) preconditioner for nodal elements on rectangular meshes as well as unstructured simplex meshes in both two and three dimensions [1,2].

For simplex meshes we consider high-order nodal spectral elements based on the electrostatic and Fekete distributions and tensoral GLL nodes in the case of rectangular grids. A low-order finite element preconditioner over a local tessellation of the element is constructed. We detail the use of AMG in this context for elliptic problems and investigate the performance. Notably, by utilizing the low-order AMG preconditioner, the solution process maintains optimal complexity for moderately high-order elements ($p=12$ in 2D and $p=10$ in 3D). Furthermore, with a modified AMG preconditioner, we are able to achieve similar performance for nonconforming discretizations such as the local discontinuous Galerkin method.