The Hodge Laplacian on Axisymmetric Domains and its Discretization

TL;DR

This paper investigates the Hodge Laplacian on axisymmetric domains using finite element methods, focusing on stability and error estimates with low-regularity functions.

Contribution

It introduces a new finite element framework with commuting projectors for axisymmetric problems, enabling stable discretizations with general data.

Findings

Established stability results for the mixed formulation.

Derived error estimates for low-regularity functions.

Developed commuting projectors applicable to weighted function spaces.

Abstract

We study the mixed formulation of the abstract Hodge Laplacian on axisymmetric domains with general data through Fourer-finite-element-methods in weighted functions spaces. Closed Hilbert complexes and commuting projectors are used through a family of finite element spaces recently introduced for general axisymmetric problems. In order to get stability results and error estimates for the discrete mixed formulation, we construct commuting projectors that can be applied to functions with low regularity.