# Higher-order meshing of implicit geometries - part II: Approximations on manifolds

## TL;DR

This paper introduces an automatic higher-order surface meshing method for PDEs on manifolds using level-set data, enabling seamless geometric to numerical analysis workflows with optimal convergence.

## Contribution

It presents a novel automatic meshing approach for PDEs on manifolds that ensures high accuracy and shape regularity without user intervention.

## Key findings

- Achieves optimal convergence rates in numerical solutions.
- Maintains shape regularity and sufficient continuity of surface elements.
- Operates with a moderate increase in condition number compared to handcrafted meshes.

## Abstract

A new concept for the higher-order accurate approximation of partial differential equations on manifolds is proposed where a surface mesh composed by higher-order elements is automatically generated based on level-set data. Thereby, it enables a completely automatic workflow from the geometric description to the numerical analysis without any user-intervention. A master level-set function defines the shape of the manifold through its zero-isosurface which is then restricted to a finite domain by additional level-set functions. It is ensured that the surface elements are sufficiently continuous and shape regular which is achieved by manipulating the background mesh. The numerical results show that optimal convergence rates are obtained with a moderate increase in the condition number compared to handcrafted surface meshes.