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