In this paper, an implicit finite element-discontinuous Galerkin method for compressible viscous and inviscid flow is developed using Newton-Krylov algorithm with the objective of increasing the accuracy and convergence rate. For inviscid flows, an artificial viscosity is implemented in sharp gradient flow regions especially at high-order cases, increasing the accuracy of the solution. Moreover, for viscous flows, the accuracy is improved by using compact discontinuous Galerkin discretization method for elliptical terms. To reduce the computing CPU time and increase the convergence rate, an iterative Krylov type preconditioned linear solver is applied. For preconditioning, restarting, Block-Jacobi and block incomplete-LU factorization are employed for solving the linear system of the Jacobian matrix. The Jacobian matrix is constructed via finite difference perturbation technique. In this context, the performance of preconditioning matrix for three types of flow regimes of inviscid subsonic, inviscid transonic and viscous laminar subsonic are studied. In addition to complete the discussions, multigrid smoother with special conditions is applied for all preconditioning matrices. To improve the solver performance for higher order discretization, a lower order solution may be used as higher orders initial condition. Therefore, a middle phase is needed to transfer calculations from low to high order discretized domain and then the final Newton phase is continued. In addition, local time stepping is implemented to improve the rate of convergence. Consequently, the presented numerical method can be used as an efficient algorithm for high-order Discontinuous Galerkin flow simulation, especially for transonic inviscid and laminar viscous flows.