A multigrid method based on the domain partitioning approach is presented in this work. A special prolongation technique is used to provide accurate boundary conditions and good initial iterate for the reduced problems on partitioned domains. When used in a high-order τ-extrapolated multigrid setting, this technique provides highly accurate solution on the finest grid on the portion of the domain where the solution is smooth. The prolongation technique is also useful in obtaining good initial guesses on the finest grid for any Newton-like multigrid method.