Fast Sweeping Methods for Eikonal Equations on Triangular Meshes

The original fast sweeping method, which is an efficient iterative method for stationary Hamilton-Jacobi equations, relies on natural ordering provided by a rectangular mesh. We propose novel ordering strategies so that the fast sweeping method can be extended efficiently and easily to any unstructured mesh. To that end we introduce multiple reference points and order all the nodes according to their $l^p$-metrics to those reference points. We show that these orderings satisfy the two most important properties underlying the fast sweeping method: (1) these orderings can cover all directions of information propagating efficiently; (2) any characteristic can be decomposed into a finite number of pieces and each piece can be covered by one of the orderings. We prove the convergence of the new algorithm. The computational complexity of the algorithm is nearly optimal in the sense that the total computational cost consists of $O(M)$ flops for iteration steps and $O(M{\rm log}M)$ flops for sorting at the predetermined initialization step which can be efficiently optimized by adopting a linear time sorting method, where $M$ is the total number of mesh points. Extensive numerical examples demonstrate that the new algorithm converges in a finite number of iterations independent of mesh size.