As we will see in this thesis, elliptic boundary value problems are the type of problem to which multigrid methods can be applied very efficiently. However, multigrid or multigrid-like methods have also been developed for many PDEs with nonelliptic features. Multigrid as an iterative linear solver The most basic way to view multigrid is to consider it as an iterative linear solver for a discrete elliptic boundary value problem. Here we assume the problem, the grid and the discretization to be given and fixed. A characteristic feature of the iterative multigrid approach is that the multigrid convergence speed is independent of the discretization mesh size h and that the number of arithmetic operations per iteration step is proportional to the number of grid points. The multigrid principle allows us to construct very efficient linear solvers, and, in that respect, the iterative approach is important and fundamental. Multigrid methods are generally accepted as being the fastest numerical methods for the solution of elliptic partial differential equations. Furthermore, they are regarded as among the fastest methods for many other problems, like other types of partial differential equations, integral equations etc. If the multigrid idea is generalized to structures other than grids, one obtains multilevel, multiscale or multiresolution methods, which can also be used successfully for very different types of problems, e.g. problems which are characterized by matrix structures, particle structures, lattice structures etc. However, the literature does not have a uniform definition of the terms multigrid, multilevel etc. The multigrid method is among the most efficient iterative methods for solving linear systems arising from discretizing elliptic differential equations. The multigrid ideas is based on two principles, error smoothing and coarse grid. The algorithm of multigrid have various components. Experience with multigrid methods shows that the choice of these components may have a strong influence on the efficiency of the resulting algorithm. On the other hand, there are no general simple result on how to choose the individual components in order to construct optimal algorithm for complicated applications. In this thesis we, first consider the multigrid methods generally, then we combine a compact high-order difference approximation with multigrid V-cycle algorithm to solve the Poisson equation with Drichlet boundary conditions. One iteration of a simple multigrid V-cycle consists of smoothing the error using a relaxation technique, solving an approximation to the smooth error equation on a coarse grid, interpolation the error correction to the fine grid and finally adding the error correctio into the approximation. This scheme along with several different ordering of grid space and projection operators, is compared with standard second-order difference formula to show improvement i accuracy. Finally, we employed a fourth-compact difference scheme with unequal mash sizes for discretiziting Possion equation. Special multigrid methods are developed to solve the resulting sparse linear systems efficiently . The multigrid methods with line Gauss-Seidel relaxation or partial semicoarsing are found to work very well in solving the fourth-compact schemes discretized Poisson equation with unequal mesh sizes. Convergence behavior of the partial semi coarsing and line Gauss-Seidel relaxation multigrid methods is examined experimentally.