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Algebraic multigrid for higher-order finite elements
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DOI：10.1016/j.jcp.2004.10.021.png)
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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. (c) 2004 Elsevier Inc. All rights reserved.
Keyword:
multigrid
algebraic multigrid
higher-order
finite elements
Poisson
Stokes
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IF:
3.8
论文数:
1.6W
被引数:
7.4W
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