#### Document Type

Post-Print

#### Publication Date

2010

#### Subjects

Multigrid methods (Numerical analysis), Electromagnetism, Symmetry (Mathematics)

#### 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 Vcycle 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.

#### DOI

10.1090/S0025-5718-2010-02384-1

#### Persistent Identifier

http://archives.pdx.edu/ds/psu/10656

#### Citation Details

Copeland, Dylan M.; Gopalakrishnan, Jay; and Oh, Minah, "Multigrid in a Weighted Space Arising from Axisymmetric Electromagnetics" (2010). *Mathematics and Statistics Faculty Publications and Presentations*. 53.

http://archives.pdx.edu/ds/psu/10656

## Description

This is an Author's Accepted Manuscript. First Published in Mathematics of Computation. April 2010, Vol. 79 Issue 272, p. 2033-2058.