In this paper, we consider the discontinuous Galerkin finite element method for the strongly nonlinear elliptic boundary value problems in a convex polygonal $$ \varOmega \subset {\mathbb R}^2.$$ Ω ⊂ R 2 . Optimal and suboptimal order pointwise error estimates in the $$W^{1,\infty }$$ W 1 , ∞ -seminorm and in the $$L^{\infty }$$ L ∞ -norm are established on a shape-regular grid under the regularity assumptions $$u\in W^{r+1,\infty }(\varOmega ), r\ge 2$$ u ∈ W r + 1 , ∞ ( Ω ) , r ≥ 2 . Moreover, we propose some two-grid algorithms for the discontinuous Galerkin method which can be thought of as some type of linearization of the nonlinear system using a solution from a coarse finite element space. With this technique, solving a nonlinear elliptic problem on the fine finite element space is reduced into solving a linear problem on the fine finite element space and solving the nonlinear elliptic problem on a coarser space. Convergence estimates in a mesh-dependent energy norm are derived to justify the efficiency of the proposed two-grid algorithms. Numerical experiments are also provided to confirm our theoretical findings.