We employ a damped Newton multigrid algorithm to solve a nonlinear system arising from a finite-difference discretization of an elliptic flame sheet problem. By selecting the generalized minimum residual method as the linear smoother for the multigrid algorithm, we conduct a series of numerical experiments to investigate the behavior and efficiency of the multigrid solver in solving the linearized systems, by choosing several preconditioners for the Krylov subspace method. It is shown that the overall efficiency of the damped Newton multigrid algorithm is highly related to the quality of the preconditioner chosen and the number of smoothing steps done on each level. ILU preconditioners based on the Jacobian pattern are found to be robust and provide efficient smoothing but at an expensive cost of storage. It is also demonstrated that the technique of mesh sequencing and multilevel correction scheme provides significant CPU saving for fine grid calculations by limiting the growth of the Krylov iterations.
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