Pointwise Error Estimates and Two-Grid Algorithms of Discontinuous Galerkin Method for Strongly Nonlinear Elliptic Problems

Abstract

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?R2. Optimal and suboptimal order pointwise error estimates in the $$W^{1,\infty }$$W1,?-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?Wr+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.