Nodal auxiliary space preconditioning for the surface de Rham complex

TL;DR

This paper introduces optimal preconditioners for discrete surface H(curl) and H(div) problems, enabling efficient computation of harmonic vector fields on complex 2D surfaces using auxiliary space techniques.

Contribution

It develops fast, user-friendly preconditioners for surface curl-curl and grad-div problems based on inverting surface Laplacians, extending auxiliary space methods to surface PDEs.

Findings

Preconditioners significantly improve iterative solver efficiency.

Numerical tests confirm robustness on 2D and 3D hypersurfaces.

Method is applicable to unstructured triangulated surfaces.

Abstract

This work develops optimal preconditioners for the discrete H(curl) and H(div) problems on two-dimensional surfaces by nodal auxiliary space preconditioning [R. Hiptmair, J. Xu: SIAM J. Numer. Anal. \textbf{45}, 2483-2509 (2007)]. In particular, on unstructured triangulated surfaces, we develop fast and user-friendly preconditioners for the edge and face element discretizations of curl-curl and grad-div problems based on inverting several discrete surface Laplacians. The proposed preconditioners lead to efficient iterative methods for computing harmonic tangential vector fields on discrete surfaces. Numerical experiments on two- and three-dimensional hypersurfaces are presented to test the performance of those surface preconditioners.