An implicit Discontinuous Galerkin scheme for conservation laws on unstructured mesh system is developed utilizing a pointwise relaxation algorithm. First, we solve the linear hyperbolic equation to examine the order of spatial accuracy and also the convergence property of the scheme. The Euler equations in two-dimensional space are then solved by the present pointwise relaxation scheme assuming a second order of spatial accuracy. We also focus on to examine the case of using a hybrid mesh system. Then we extend the present method to solve the Navier-Stokes equations. It is shown that the present pointwise relaxation method can maintain numerical stability even with very large CFL numbers, and can converge to machine zero.