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Mathematics of Computation

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Multigrid in a weighted space arising from axisymmetric electromagnetics


Authors: Dylan M. Copeland, Jayadeep Gopalakrishnan and Minah Oh
Journal: Math. Comp. 79 (2010), 2033-2058
MSC (2010): Primary 65M55, 65N55, 65F10, 65N30, 78M10
Published electronically: May 24, 2010
MathSciNet review: 2684354
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Abstract: Consider the space of two-dimensional vector functions whose components and curl are square integrable with respect to the degenerate weight given by the radial variable. This space arises naturally when modeling electromagnetic problems under axial symmetry and performing a dimension reduction via cylindrical coordinates. We prove that if the original three-dimensional domain is convex, then the multigrid V-cycle applied to the inner product in this space converges, provided certain modern smoothers are used. For the convergence analysis, we first prove several intermediate results, e.g., the approximation properties of a commuting projector in weighted norms, and a superconvergence estimate for a dual mixed method in weighted spaces. The uniformity of the multigrid convergence rate with respect to mesh size is then established theoretically and illustrated through numerical experiments.


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Additional Information

Dylan M. Copeland
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: copeland@math.tamu.edu

Jayadeep Gopalakrishnan
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611–8105
Email: jayg@ufl.edu

Minah Oh
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611–8105
Email: oh@ufl.edu

DOI: https://doi.org/10.1090/S0025-5718-2010-02384-1
Keywords: Multigrid, axisymmetric, weighted Sobolev spaces, Maxwell equations, V-cycle, duality, superconvergence, mixed method, finite element
Received by editor(s): November 17, 2008
Received by editor(s) in revised form: June 11, 2009
Published electronically: May 24, 2010
Additional Notes: This work was supported in part by the National Science Foundation under grants DMS-0713833 and SCREMS-0619080.
Article copyright: © Copyright 2010 American Mathematical Society