Multigrid Methods For Several Kinds Of Nonlinear Problems

Nonlinear elliptic problems arise from many fields, such as engineering, mech-anism, physics, optimal control theory and so on. It becomes more and moreimportant in modern science and technology. Hence, it is meaningful to study thenumerical methods for solving nonlinear elliptic problems especially the effcientnumerical methods for large scale problems. People hope to obtain the solutionwith required accuracy of n-dimensional system through O(n) times of multiplica-tion and division operations. Multigrid method achieves this goal for the first timeand becomes the most effective method to solve large scale problems. In the pastdecades, a systematic theory on multigrid method has been developed, and manyfruits have been taken in solving nonlinear problems. In this thesis, we shall dis-cuss the multigrid methods for semilinear elliptic problems and nonsmooth ellipticproblems.For the usual semilinear elliptic problems, the semilinear term is always as-sumed to be suficiently smooth or C~2- continuous. It is very important to decreasethe smoothness of the semilinear term for semilinear elliptic problems. In Chapter2, we consider the numerical solution of a kind of semilinear elliptic problems, inwhich the derivative of the semilinear term is locally H¨older continuous. We firstinvestigate the standard finite element error estimates of this kind of problems. Wethen solve the corresponding discrete problems using cascadic multigrid method.We prove that the algorithm has the optimal order of convergence in energy normand quasi-optimal computational complexity. Numerical results show that themethod is effcient.In Chapter 3, we propose a lumped mass cascadic multigrid method for semi-linear elliptic equations. We first investigate the L~2-error estimate for the lumpedmass finite element method. There are two advantages of the lumped mass dis-cretization. From the computational point of view, one advantage of such dis-cretization is that the Jacobian of the nonlinear function involved in the discretesystem is easy to compute since the function is the sum of a linear function and adiagonal nonlinear function; Another advantage is that one can construct iterativeschemes that posses monotone convergence property to solve the correspondingdiscrete problem. On the basis of the finite element error estimates, we provethe optimality of the proposed multigrid method. The numerical results show theeffciency of the proposed algorithm. In Chapter 4, we investigate a nonsmooth Newton multigrid method for non-smooth elliptic equations. We first use finite element method to discrete a non-smooth elliptic equation and present some error estimates. Usually, nonsmoothNewton-like method is applied to solve the discrete problem. Since the Newton'sequations have a very bad conditioner when the mesh-size is finer and it is noteasy to solve. And when the scale of the discrete problem is large, it takes a greatdeal of computational work to solve the subproblem. To overcome these draw-backs, multigrid technique can be used to solve the subproblem at each Newtonstep. Under appropriate conditions, we prove the mesh-independent convergenceand (nearly) optimal property of the proposed algorithm. Numerical results areillustrated to confirm the error estimates we obtained and the effciency of thenonsmooth Newton-like multigrid method. Especially, if the mesh-size h becomesmuch smaller, the method can save substantial computational work than the usualnonsmooth Newton methods or active set method.In Chapter 5, we propose a smoothing Newton multigrid method to solve thenonsmooth elliptic problem. The merits of smoothing methods and smoothingNewton methods are global convergence. Furthermore, when the scale of the dis-crete problem is large, it takes a great deal of computational work to solve thesubproblem exactly or iteratively. So, the multigrid technique is used in solvingthe corresponding subproblems. Numerical experiments prove that the proposedalgorithm is convergent for difierent smoothing functions and show the effciencyof the proposed algorithm.