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A new approach to the analysis of axisymmetric problems. (English) Zbl 1303.65038
Summary: This paper builds a new framework to analyse axisymmetric problems through differential forms and exterior calculus. We construct a new weighted $$L^{2}$$ 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.

##### MSC:
 65J05 General theory of numerical analysis in abstract spaces 58J05 Elliptic equations on manifolds, general theory 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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