A dynamic domain decomposition for the eikonal-diffusion equation

We propose a parallel algorithm for the numerical solution of the eikonal-diffusion equation, by means of a dynamic domain decomposition technique. The new method is an extension of the patchy domain decomposition method presented in [5] for first order Hamilton-Jacobi-Bellman equations. Using the connection with stochastic optimal control theory, the semi-Lagrangian scheme underlying the original method is modified in order to deal with (possibly degenerate) diffusion. We show that under suitable relations between the discretization parameters and the diffusion coefficient, the parallel computation on the proposed dynamic decomposition can be faster than that on a static decomposition. Some numerical tests in dimension two are presented, in order to show the features of the proposed method.