Uniform convergence of V-cycle multigrid algorithms for two-dimensional fractional Feynman-Kac equation

摘要

In this paper we derive new uniform convergence estimates for the V-cycle MGMapplied to symmetric positive definite Toeplitz block tridiagonal matrices, byalso discussing few connections with previous results. More concretely, thecontributions of this paper are as follows: (1) It tackles the Toeplitz systemsdirectly for the elliptic PDEs. (2) Simple (traditional) restriction operatorand prolongation operator are employed in order to handle general Toeplitzsystems at each level of the recursion. Such a technique is then applied tosystems of algebraic equations generated by the difference scheme of thetwo-dimensional fractional Feynman-Kac equation, which describes the jointprobability density function of non-Brownian motion. In particular, we considerthe two coarsening strategies, i.e., doubling the mesh size (geometric MGM) andGalerkin approach (algebraic MGM), which lead to the distinct coarseningstiffness matrices in the general case: however, several numerical experimentsshow that the two algorithms produce almost the same error behaviour.