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In this paper, we will construct a discrete de Rham complex with commuting interpolation operators for Fourier finite element methods in axisymmetric domains. To our knowledge, this will be the first construction of a discrete de Rham complex and commuting projectors that can be applied to axisymmetric problems with general data. [ABSTRACT FROM AUTHOR]

This paper builds a new framework to analyse axisymmetric problems through differential forms and exterior calculus. We construct a new weighted L2 de Rham complex that arises when analysing axi-symmetric problems. By constructing its dual complex, we obtain a Hodge decomposition and a Poincaré inequality in weighted spaces, and the well-posedness of the weighted mixed Hodge Laplacian problem. The stability and convergence of the discrete weighted mixed Hodge Laplacian problem are also achieved by using bounded cochain projections. [ABSTRACT FROM AUTHOR]

In this paper, we will construct and analyze a multigrid algorithm that can be applied to weighted H (div) problems on a two-dimensional domain. These problems arise after performing a dimension reduction to a three-dimensional axisymmetric H (div) problem. We will use recently developed Fourier finite element spaces that can be applied to axisymmetric H (div) problems with general data. We prove that if the axisymmetric domain is convex, then the multigrid V-cycle with modern smoothers will converge uniformly with respect to the meshsize. [ABSTRACT FROM AUTHOR]