Algebraic Multigrid (AMG) for Higher-Order Finite Elements

Two related approaches for solving linear systems that arise from a higher-order nite element discretization of elliptic partial dieren tial equations are described. The rst 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 signican t impact on the performance of the solver, results indicate that AMG is capable of solving operators from both Poisson’s equation and a rst-order system least-squares (FOSLS) formulation of Stoke’s equation in a scalable manner, nearly independent of basis order, p, for 3 < p 8. The second approach incorporates preconditioning based on a bilinear nite element mesh overlaying the entire set of degrees of freedom in the higher-order scheme. AMG is applied to the operator based on bilinear nite elements 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. Although the total iteration count is slightly higher than using AMG accelerated by CG directly on the high-order operator, the preconditioned approach has the advantage of a straightforward matrix-free implementation of the high-order operator, thereby avoiding typically large computational and storage costs.