Abstarct: The finite element method is a very well-known and efficient scheme to the discretization of PDE equations. Among some discretization tools, we pick this one to discretize the heat equation in two and three dimensions. It is more involved and more general than that group of finite difference methods since the mass matrix must be considered. The resulted sparse matrix should be solved efficiently by special schemes such as the general group of multigrid methods. We refer to the well-known v-cycle, w-cycle which usually have been applied and analysed on the elliptic equations by many researchers. So, in this thesis, we work on the parabolic equations to keep the generality of the study. As some results of our work, we can refer to presenting a new two grid, investigation of the v- and w-cycles on rectangular and general triangular grids in two and three dimensions. moreover, a nonlinear system of control equations is solved by FAS multigrid methods. comprehensively study of different convergence analysis and extending well them is one of the big advantages of this thesis.