Compact Fourier Analysis for Designing Multigrid Methods

The convergence of Multigrid methods can be analyzed based on a Fourier analysis of the method or by proving certain inequalities that have to be fulfilled by the smoother and by the coarse grid correction separately. Here, we analyze the Multigrid method for the constant coefficient Poisson equation with a compact Fourier analysis using the formalism of multilevel Toeplitz matrices and their generating functions or symbols. The Fourier analysis is applied for determining the smoothing factor and the overall error of the combined smoothing and coarse grid correction error reduction of a Twogrid step by representing the Twogrid step explicitly by a symbol. If the effects of the smoothing correction and the coarse grid correction are orthogonal to each other, then in a Twogrid step the error is removed in one step and the Twogrid method can be considered as a direct solver. If the coarse linear system is identical to the original matrix, then the same projection and smoother again make t...

• Type: Journal
• Year: 2008
• Where: SIAMSC
• Authors: Thomas K. Huckle
• Keywords: Coarse Grid Correction, Fourier Analysis, Multigrid Method, SIAMSC 2008