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

TL;DR

This paper establishes uniform convergence estimates for V-cycle multigrid methods applied to symmetric positive definite Toeplitz matrices, specifically for the 2D fractional Feynman-Kac equation, enhancing understanding of multigrid efficiency for these PDEs.

Contribution

It provides new uniform convergence results for multigrid algorithms directly applied to Toeplitz systems from 2D fractional PDEs, using simple operators for general systems.

Findings

Multigrid methods show similar error behavior for geometric and algebraic coarsening.

The approach effectively handles Toeplitz systems from fractional Feynman-Kac equations.

Numerical experiments confirm the theoretical convergence estimates.

Abstract

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.