Some Multigrid Methods For Solving The P-Laplacian Equations

In this article, a new multigrid method for solving p-Laplacian equations is proposed based on the existing multigrid method: FAS multigrid method and Cascade multigrid method. This method is a combination of Cascade method and a new method named "back" method. It gathers the ideas of classic multigred method and Cascadic multigrid method. It fastens and improves the precision of Cascade method by using its coarser grid correction. And it holds the right side of the original equation, so that it ensures the property of the correction equation will be similar to the original equation.Since the solution of the problem is equivalent to minimize a strictly convex functional, we use three kinds of unconstraint nonlinear optimization method in these multigrid methods, a.e. Polack-Ribiere conjugate gradient method, Hooke-Jeeves pattern search method and a SSC gradient method without line search. The advantage of doing so is that we do not have to calculate derivatives of operators, for it is very difficult.Polack-Ribiere conjugate gradient method is not efficient and even not convergent when (p+l)/(p-l) or p is large and not so good initial value is given. So in this paper, we use a much more robust algorithm, Hooke-Jeeves pattern search method, to solve the problem on the coarse grid, and the faster method: Polack-Ribiere conjugate gradient method or SSC gradient method on the finer grid.In this paper, some numerical results are given for different p in one dimension and two dimension, and base on the result we give the comparison and analysis of the efficiency of different multigrid methods and they used different smoother,and prove the valid of the Cascade-back method.