Higher-order meshing of implicit geometries - part II: Approximations on manifolds
This work automates the mesh generation process for higher-order surface FEM, eliminating user intervention for geometric descriptions, which is important for computational geometry and numerical analysis.
The paper proposes a fully automatic method for generating higher-order surface meshes from level-set data to approximate PDEs on manifolds, achieving optimal convergence rates with only a moderate increase in condition number compared to handcrafted meshes.
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.