Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Comparison of Multigrid Algorithms for High-order Continuous Finite
  Element Discretizations   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 (SSOR) 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 $h$- and $p$-multigrid work well; the first-order
approximation is less efficient. For constant coefficients, all smoothers work
well. For variable coefficients, Chebyshev and SSOR smoothing outperforms
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 to
using multigrid as a solver.

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The categories of this paper are: gr-qc
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The categories of this paper are: mtrl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-lat
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The categories of this paper are: astro-ph