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The convergence of V-cycle multigrid algorithms for axisymmetric Laplace and Maxwell equations

Authors: Jayadeep Gopalakrishnan and Joseph E. Pasciak
Journal: Math. Comp. 75 (2006), 1697-1719
MSC (2000): Primary 65F10, 65M55, 65N55, 65N30, 49N60, 74G15, 35Q60
Published electronically: July 6, 2006
MathSciNet review: 2240631
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Abstract: We investigate some simple finite element discretizations for the axisymmetric Laplace equation and the azimuthal component of the axisymmetric Maxwell equations as well as multigrid algorithms for these discretizations. Our analysis is targeted at simple model problems and our main result is that the standard V-cycle with point smoothing converges at a rate independent of the number of unknowns. This is contrary to suggestions in the existing literature that line relaxations and semicoarsening are needed in multigrid algorithms to overcome difficulties caused by the singularities in the axisymmetric Maxwell problems. Our multigrid analysis proceeds by applying the well known regularity based multigrid theory. In order to apply this theory, we prove regularity results for the axisymmetric Laplace and Maxwell equations in certain weighted Sobolev spaces. These, together with some new finite element error estimates in certain weighted Sobolev norms, are the main ingredients of our analysis.

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Additional Information

Jayadeep Gopalakrishnan
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611–8105

Joseph E. Pasciak
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843–3368

Keywords: Multigrid, axisymmetric, Laplace equation, Maxwell equations, V-cycle, Jacobi, Gauss--Seidel, regularity, bilinear, finite element
Received by editor(s): May 20, 2004
Received by editor(s) in revised form: September 16, 2005
Published electronically: July 6, 2006
Additional Notes: This work was supported in part by NSF grant numbers DMS-0410030 and DMS-0311902. We also gratefully acknowledge support from ICES, The University of Texas at Austin.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.