### Abstract

Two related approaches for solving linear systems that arise from a higher-order finite element discretization of elliptic partial differential equations are described. 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, for 3 < p ≤ 8. 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 bilinear finite 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.

Original language | English (US) |
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Pages (from-to) | 520-532 |

Number of pages | 13 |

Journal | Journal of Computational Physics |

Volume | 204 |

Issue number | 2 |

DOIs | |

State | Published - Apr 10 2005 |

Externally published | Yes |

### Keywords

- Algebraic multigrid
- Finite elements
- Higher-order
- Multigrid
- Poisson
- Stokes

### ASJC Scopus subject areas

- Numerical Analysis
- Modeling and Simulation
- Physics and Astronomy (miscellaneous)
- Physics and Astronomy(all)
- Computer Science Applications
- Computational Mathematics
- Applied Mathematics

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## Cite this

*Journal of Computational Physics*,

*204*(2), 520-532. https://doi.org/10.1016/j.jcp.2004.10.021