A note on grid transfer operators for multigrid methods

TL;DR

This paper demonstrates the equivalence of two approaches for analyzing grid transfer operators in multigrid methods, extending the theory to Toeplitz matrices and linking wavelets with multigrid techniques.

Contribution

It shows the equivalence between LFA and Toeplitz symbol analysis for grid transfer operators in elliptic PDEs, enabling broader application and new operator classes.

Findings

Equivalence of LFA and Toeplitz symbol approaches for grid transfer operators.

Introduction of B-spline based grid transfer operators with specific properties.

Numerical experiments confirming the theoretical results.

Abstract

The Local Fourier analysis (LFA) is a classic tool to prove convergence theorems for multigrid methods (MGMs). In particular, we are interested in optimality that is a convergence speed independent of the size of the involved matrices. For elliptic partial differential equations (PDEs), a well known optimality result requires that the sum of the orders of the grid transfer operators is not lower than the order of the PDE to solve. Analogously, when dealing with MGMs for Toeplitz matrices in the literature an optimality condition on the position and on the order of the zeros of the symbols of the grid transfer operators has been found. In this work we show that in the case of elliptic PDEs with constant coefficients, the two different approaches lead to an equivalent condition. We argue that the analysis for Toeplitz matrices is an algebraic generalization of the LFA, which allows to deal not only with differential problems but also for instance with integral problems. The equivalence of the two approaches gives the possibility of using grid transfer operators with different orders also for MGMs for Toeplitz matrices. We give also a class of grid transfer operators related to the B-spline's refinement equation and we study their geometric properties. This analysis suggests further links between wavelets and multigrid methods. A numerical experimentation confirms the correctness of the proposed analysis.