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Weak discrete maximum principle and \(L^\infty \) analysis of the DG method for the Poisson equation on a polygonal domain

  • Yuki ChibaEmail author
  • Norikazu Saito
Original Paper
  • 12 Downloads

Abstract

We derive several \(L^\infty \) error estimates for the symmetric interior penalty discontinuous Galerkin method applied to the Poisson equation in a two-dimensional polygonal domain. Both local and global estimates are examined. The weak maximum principle (WMP) for the discrete harmonic function is also established. We prove our \(L^\infty \) estimates using this WMP and several \(W^{2,p}\) and \(W^{1,1}\) estimates for the Poisson equation. Numerical examples to validate our results are also presented.

Keywords

Discontinuous Galerkin method Pointwise error estimate Maximum principle 

Mathematics Subject Classification

65N15 65N30 

Notes

Acknowledgements

We thank the anonymous reviewer for his/her valuable comments and suggestions to improve the quality of the paper. Yuki Chiba was supported by Program for Leading Graduate Schools, MEXT, Japan. Norikazu Saito was supported by JST CREST Grant number JPMJCR15D1, Japan, and JSPS KAKENHI Grant number 15H03635, Japan.

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Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

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