Cascadic Multigrid For Semilinear Parabolic Problem

The cascadic multigrid method has been shown to be one of the most efficient iterative techniques for solving large scale boundary value problems.The main advantage of the method is coarse-grid-correction free,and as a result it can be viewed as a one-way multigrid method. Gisela Timmermann proposed a cascadic multigrid for a semilinear elliptic problem. On the coarest grid the nonlinear equations arising from linear finite element discretizations are solved by Newton's method.On the fine grid the Newton's equations is solved by the cascadic multigrid method.In this paper we extend the cascadic multigrid method to semilinear parabolic problems. It has been proved that the method has optimal convergence order of the error in energy norm, and has the optimal or quasi-optimal computation complexity.We construct cascadic multigrid method for a model problem-a semilinear parabolic problem. On the coarest grid the nonlinear equations arising from linear finite element discretizations are solved by Newton's method.On the fine grid the Newton's equations is solved by the cascadic multigrid method.At first we use Richardson iteration as smoothing operator and prove the method has optimal convergence order for the error in the energy norm to 2-D semilinear parabolic problem. Then we use conjugate gradient(CG) as smoothers and prove this method has optimal convergence order of the error in the energy norm also. For these two cases, we analyse the computational complexity.The optimality or quasi-optimalityof the computation of the method is shown.This fact displays that for cascadic multigrid the computational work in semilinear case is almost the same as that in linear case.Finally a numerical experiment is given to the effectiveness of the method.