A GENERALIZED FAST MARCHING METHOD ON UNSTRUCTURED TRIANGULAR MESHES

In this paper we extend the generalized fast marching method (GFMM) presented in [E. Carlini et al., SIAM J. Numer. Anal., 46 (2008), pp. 2920–2952] to unstructured meshes. The GFMM generalizes the classical fast marching method, in the sense that it can be applied to propagate interfaces with time-dependent and changing sign velocity. The main motivation for this extension is that in many areas, such as in fluid dynamics, the method for tracking the motion of an interface in time has to be coupled with other solvers, based on finite elements or finite volumes, typically constructed on triangular grids and on domains with complex geometries. We prove an abstract convergence result for the two-dimensional (2D) case, which requires only some properties of the mesh and of the local solvers, used to compute the solution on a narrow band. In particular, this result applies to a class of monotone schemes for 2D problems on triangular grids with acute angles. Finally, some numerical tests in two dimensions illustrate the effectiveness and the main features of the method.