# Finite element procedures for computing normals and mean curvature on triangulated surfaces and their use for mesh refinement

## TL;DR

This paper introduces finite element methods with stabilization techniques for accurately computing normals and mean curvature on triangulated surfaces, enabling adaptive mesh refinement and surface reconstruction.

## Contribution

It presents a stabilized finite element approach for mean curvature and normals, improving convergence and enabling adaptive mesh refinement on triangulated surfaces.

## Key findings

- Achieves first order L^2-convergence for mean curvature vector.
- Provides a method for continuous normal recovery using L^2-projections.
- Demonstrates improved surface reconstruction and mesh refinement results.

## Abstract

In this paper we consider finite element approaches to computing the mean curvature vector and normal at the vertices of piecewise linear triangulated surfaces. In particular, we adopt a stabilization technique which allows for first order $L^2$-convergence of the mean curvature vector and apply this stabilization technique also to the computation of continuous, recovered, normals using $L^2$-projections of the piecewise constant face normals. Finally, we use our projected normals to define an adaptive mesh refinement approach to geometry resolution where we also employ spline techniques to reconstruct the surface before refinement. We compare or results to previously proposed approaches.