Efficient mesh optimization schemes based on Optimal Delaunay Triangulations

In this paper, several mesh optimization schemes based on Optimal Delaunay Triangulations are developed. High-quality meshes are obtained by minimizing the interpolation error in the weighted L1 norm. Our schemes are divided into classes of local and global schemes. For local schemes, several old and new schemes, known as mesh smoothing, are derived from our approach. For global schemes, a graph Laplacian is used in a modified Newton iteration to speed up the local approach. Our work provides a mathematical foundation for a number of mesh smoothing schemes often used in practice, and leads to a new global mesh optimization scheme. Numerical experiments indicate that our methods can produce well-shaped triangulations in a robust and efficient way.