Multigrid method for symmetric Toeplitz block tridiagonal matrix: Convergence analysis and application
Minghua Chen, Weihua Deng · arXiv (Cornell University) · 2016
The V-cycle multigrid method can significantly lower the computational cost, yet their uniform convergence properties have not been systematically examined, even for the classical parabolic partial differential equations. The main contribution of this paper is that by improving the framework of convergence estimates for multigrid method [Math. Comp., 49 (1987), pp. 311--329], we derive the uniform convergence estimates of the V-cycle multigrid method for symmetric positive definite Toeplitz block tridiagonal matrix. Then it is used to the algebraic equation 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, for the multigrid method, there exists two coarsening strategies, i.e., doubling the mesh size and Galerkin approach, which lead to the different coarsening stiffness matrices in the general case, but the numerical experiments show that they have almost the same error results.