Wavelet based multigrid methods for linear and nonlinear elliptic partial differential equations

Multigrid method is a well known computational technique for accelerating convergence to the steady state for flow problems and has proved to be very successful for solving elliptic equations. Fedorenko [6] was the first to formulate a multigrid algorithm for the standard five point finite difference discretization of the Poisson equation on a square. The first practical results using the multigrid method were reported in the pioneering papers by Brandt [1,2]. Independently, Hackbush [11] discovered the multigrid method and laid firm mathematical foundations and provided reliable multigrid schemes. Since then many variants of the method have evolved to solve different partial differential equations with different intergrid operators (interpolation and restriction). The main reason for these techniques to be well suited for elliptic equations is the dampening property which reduce both high and low frequency errors. A routine Fourier analysis demonstrates that most of the commonly used solvers effectively damp only the high frequency components of the errors. Since a low frequency component of the error of a fine mesh appears as a high frequency component on a coarser one, it proves effective to solve the problem on a sequence of fine and coarse grids. Both high and low frequency components of the error are then damped equally and if enough grids are available, only a few iterations will be required to produce a converged solution.