The solution of the compressible Euler and Navier-Stokes equations via an upwind finite volume scheme is obtained. For the inviscid fluxes, the monotone upstream-centered scheme for conservation laws (MUSCL) has been incorporated into a Riemann solver. The MUSCL scheme is used for the unfactored implicit equations that are solved by a Newton form, and relaxation is performed via Gauss-Seidel relaxation technique. The solution on the fine grid is obtained by iterating first on a sequence of coarser grids and then interpolating the solution up to the next refined grid. Since the distribution of the numerical error is nonuniform, the local solution of the equations can be obtained in regions where the numerical errors are large. The construction of the partial meshes, in which the iterations will be continued, is determined by an adaptive procedure taking into account some convergence criteria. Reduction of the computational work units for two-dimensional problems is obtained via the local adaptive mesh solution which is expected to be more effective in three-dimensional complex flow computations.