An Arbitrary-Order Discontinuous Galerkin Method with One Unknown Per Element

  • Ruo Li
  • Pingbing MingEmail author
  • Ziyuan Sun
  • Zhijian Yang


We propose an arbitrary-order discontinuous Galerkin method for second-order elliptic problem on general polygonal mesh with only one degree of freedom per element. This is achieved by locally solving a discrete least-squares over a neighboring element patch. Under a geometrical condition on the element patch, we prove an optimal a priori error estimates in the energy norm and in the \(\hbox {L}^2\) norm. The accuracy and the efficiency of the method up to order six on several polygonal meshes are illustrated by a set of benchmark problems.


Least-squares reconstruction Discontinuous Galerkin method Elliptic problem 

Mathematics Subject Classification

Primary 65N30 49N45 Secondary 74K20 



The authors would like to thank Dr. Fengyang Tang for his help in the earlier stage of the present work, and the authors would like to thank the anonymous referees for the constructive comments that improve the paper. Funding was provided by National Natural Science Foundation of China (Grant Nos. 11425106, 91630313, 91630313, 11671312)


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Authors and Affiliations

  • Ruo Li
    • 1
  • Pingbing Ming
    • 2
    • 3
    Email author
  • Ziyuan Sun
    • 4
  • Zhijian Yang
    • 5
  1. 1.CAPT, LMAM and School of Mathematical SciencesPeking UniversityBeijingPeople’s Republic of China
  2. 2.LSEC, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China
  3. 3.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingPeople’s Republic of China
  4. 4.School of Mathematical SciencesPeking UniversityBeijingPeople’s Republic of China
  5. 5.School of Mathematics and StatisticsWuhan UniversityWuhanPeople’s Republic of China

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