Multigrid with FFT smoother for a simplified 2D frictional contact problem

This paper aims to develop a fast multigrid (MG) solver for a Fredholm integral equation of the first kind, arising from the 2D elastic frictional contact problem. After discretization on a rectangular contact area, the integral equation gives rise to a linear system with the coefficient matrix being dense, symmetric positive definite and Toeplitz. A so-called fast Fourier transform (FFT) smoother is proposed. This is based on a preconditioner M that approximates the inverse of the original coefficient matrix, and that is determined using the FFT technique. The iterates are then updated by Richardson iteration: adding the current residuals preconditioned with the Toeplitz preconditioner M . The FFT smoother significantly reduces most components of the error but enlarges several smooth components. This causes divergence of the MG method. Two approaches are studied to remedy this feature: subdomain deflation (SD) and row sum modification (RSM). MG with the FFT + RSM smoother appears to be more efficient than using the FFT + SD smoother. Moreover, the FFT + RSM smoother can be applied as an efficient iterative solver itself. The two methods related to RSM also show rapid convergence in a test with a wavy surface, where the Toeplitz structure is lost. Copyright © 2014 John Wiley & Sons, Ltd.