In this dissertation, multigrid methods have been investigated for solving certain classes of obstacle problems. In the first part of this thesis, we investigated the solution of elliptic quasi-variational inequalities arising in discretization by the finite element method, in this case, we have chosen a linear right-hand side and the variational form associated with linear operator. For these problems, we have proposed a standard multigrid approach for solving a linear system obtained. Otherwise, we have proposed a nonlinear multigrid method for elliptic quasi-variational inequalities with nonlinear source terms and nonlinear variational form.
The L_?-norm convergence of these two approaches has been constructed which demonstrates that the multigrid method has a contraction number with respect to the L_?-norm. Numerical results which demonstrate the high efficiency of these methods are given for a quasi-variational inequality arising from impulse control problem on a domain with nonpolygonal boundaries. From these numerical results, we have seen that the multigrid method proves to be more efficient than the other iterative methods.