Analysis of the finite element method for the Laplace--Beltrami equation on surfaces with regions of high curvature using graded meshes
For researchers in numerical analysis and computational geometry, this provides rigorous error control for surface PDEs on complex geometries, though the approach is incremental.
The paper derives error estimates for finite element solutions of the Laplace-Beltrami equation on surfaces with high curvature regions, showing that graded meshes yield error bounds independent of curvature magnitude. Numerical experiments confirm the theory.
We derive error estimates for the piecewise linear finite element approximation of the Laplace--Beltrami operator on a bounded, orientable, $C^3$, surface without boundary on general shape regular meshes. As an application, we consider a problem where the domain is split into two regions: one which has relatively high curvature and one that has low curvature. Using a graded mesh we prove error estimates that do not depend on the curvature on the high curvature region. Numerical experiments are provided.