In this paper, we consider the discontinuous Galerkin finite element method for the strongly nonlinear elliptic boundary value problems in a convex polygonal (Formula presented.) Optimal and suboptimal order pointwise error estimates in the (Formula presented.) -seminorm and in the (Formula presented.) -norm are established on a shape-regular grid under the regularity assumptions (Formula presented.). 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.
- Discontinuous Galerkin methods
- Nonlinear problems
- Pointwise error estimates
- Two-grid algorithms
ASJC Scopus subject areas
- Theoretical Computer Science
- Computational Theory and Mathematics