Fast auxiliary space preconditioners on surfaces

TL;DR

This paper introduces efficient multilevel preconditioners for the Laplace--Beltrami operator on hypersurfaces, enhancing computational performance for discretized surface PDEs using the FASP framework.

Contribution

It develops and analyzes uniform multilevel preconditioners for surface Laplace--Beltrami problems discretized by various finite element methods within the FASP framework.

Findings

Preconditioners significantly improve convergence rates.

Numerical experiments confirm efficiency on 2D and 3D surfaces.

Applicable to semi-definite surface PDEs.

Abstract

This work presents uniform preconditioners for the discrete Laplace--Beltrami operator on hypersurfaces. In particular, within the framework of fast auxiliary space preconditioning (FASP), we develop efficient and user-friendly multilevel preconditioners for the Laplace--Beltrami type equation discretized by Lagrange, nonconforming linear, and discontinuous Galerkin elements. The analysis applies to semi-definite problems on a closed surface. Numerical experiments on 2d surfaces and 3d hypersurfaces are presented to illustrate the efficiency of the proposed preconditioners.