Title: Comparison of multigrid algorithms for high‐order continuous finite element discretizations

Abstract

Summary We present a comparison of different multigrid approaches for the solution of systems arising from high‐order continuous finite element discretizations of elliptic partial differential equations on complex geometries. We consider the pointwise Jacobi, the Chebyshev‐accelerated Jacobi, and the symmetric successive over‐relaxation smoothers, as well as elementwise block Jacobi smoothing. Three approaches for the multigrid hierarchy are compared: (1) high‐order h ‐multigrid, which uses high‐order interpolation and restriction between geometrically coarsened meshes; (2) p ‐multigrid, in which the polynomial order is reduced while the mesh remains unchanged, and the interpolation and restriction incorporate the different‐order basis functions; and (3) a first‐order approximation multigrid preconditioner constructed using the nodes of the high‐order discretization. This latter approach is often combined with algebraic multigrid for the low‐order operator and is attractive for high‐order discretizations on unstructured meshes, where geometric coarsening is difficult. Based on a simple performance model, we compare the computational cost of the different approaches. Using scalar test problems in two and three dimensions with constant and varying coefficients, we compare the performance of the different multigrid approaches for polynomial orders up to 16. Overall, both h ‐multigrid and p ‐multigrid work well; the first‐order approximation is less efficient. For constantmore » coefficients, all smoothers work well. For variable coefficients, Chebyshev and symmetric successive over‐relaxation smoothing outperform Jacobi smoothing. While all of the tested methods converge in a mesh‐independent number of iterations, none of them behaves completely independent of the polynomial order. When multigrid is used as a preconditioner in a Krylov method, the iteration number decreases significantly compared with using multigrid as a solver. Copyright © 2015 John Wiley & Sons, Ltd.« less

Authors:

Sundar, Hari ^[1]; Stadler, Georg ^[2]; Biros, George ^[3]

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+ Show Author Affiliations

1. School of Computing University of Utah Salt Lake City UT USA, Institute for Computational Engineering and Sciences The University of Texas Austin TX USA
2. Courant Institute of Mathematical Sciences New York University New York NY USA, Institute for Computational Engineering and Sciences The University of Texas Austin TX USA
3. Institute for Computational Engineering and Sciences The University of Texas Austin TX USA, Department of Mechanical Engineering The University of Texas Austin TX USA

Publication Date:

Thu Apr 02 00:00:00 EDT 2015

Sponsoring Org.:

USDOE

OSTI Identifier:

1401838

Resource Type:

Publisher's Accepted Manuscript

Journal Name:

Numerical Linear Algebra with Applications

Additional Journal Information:

Journal Name: Numerical Linear Algebra with Applications Journal Volume: 22 Journal Issue: 4; Journal ID: ISSN 1070-5325

Publisher:

Wiley Blackwell (John Wiley & Sons)

Country of Publication:

United Kingdom

Language:

English