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

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Abstract

In this paper, we consider the discontinuous Galerkin finite element method for the strongly nonlinear elliptic boundary value problems in a convex polygonal \( \varOmega \subset {\mathbb R}^2.\) Optimal and suboptimal order pointwise error estimates in the \(W^{1,\infty }\)-seminorm and in the \(L^{\infty }\)-norm are established on a shape-regular grid under the regularity assumptions \(u\in W^{r+1,\infty }(\varOmega ), r\ge 2\). 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.

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Acknowledgments

The authors wish to express their deepest gratitude to the anonymous referees who generously shared their insight and perspectives on the subject of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Yantai University, Yantai, Shandong, 264005, People’s Republic of China

    Chunjia Bi

  2. Department of Mathematics, Tongji University, Shanghai, 200092, People’s Republic of China

    Cheng Wang

  3. Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, People’s Republic of China

    Yanping Lin

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  1. Chunjia Bi
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  2. Cheng Wang
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  3. Yanping Lin
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Correspondence to Cheng Wang.

Additional information

The research of C. Bi was partially supported by Shandong Province Natural Science Foundation Grant ZR2014AM003. The research of C. Wang was partially supported by National Science Foundation of China Grant 11101311. The research of Y. Lin was partially supported by GRF 15301714 of HKSAR.

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Bi, C., Wang, C. & Lin, Y. Pointwise Error Estimates and Two-Grid Algorithms of Discontinuous Galerkin Method for Strongly Nonlinear Elliptic Problems. J Sci Comput 67, 153–175 (2016). https://doi.org/10.1007/s10915-015-0072-x

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