Two-Grid Algorithms For Semilinear Equations

Two-grid algorithms are presented in this paper for the approximations of semilinear problems. Generally, to avoid time step constrains, it's often preper-able to solve semilinear parabolic equations implicitly in time. However, for the fine grids, the resulting large systems of nonlinear equation can be costly to solve. In order to decrease the amount of work, we consider two-gird algorithms based on the finite difference method. By the proposed techniques, we can use linearized method to solve the semilinear parabolic problems and there is no limit on the mesh ratio. For the semilinear elliptic equations, we get the solution on the coarse grid by using Newton iteration and correct the rough approximation on the fine grid. A remarkable fact is that any order accuracy in coarse grid mesh size can be obtained for the approximation of solution if repeating last steps of the algorithms.This paper is divided into the following two chapters.In Chapter 1, we discuss two-grid finite difference algorithms for the following semilinear parabolic equationdefined on a rectangular domain.we assume that f is twice continuously differentiable andwhere k₁ and k₂ are positive constants independent of mesh parameters .The main process of two-grid algorithms is as follows. We apply two different spaces G_H and G_h defined respectively on one coarse grid H and one fine grid...