Interior Penalty Discontinuous Galerkin Methods for Second Order Linear Non-divergence Form Elliptic PDEs

Xiaobing Feng, Michael Neilan, Stefan Schnake

Research output: Contribution to journalArticlepeer-review

18 Scopus citations

Abstract

This paper develops interior penalty discontinuous Galerkin (IP-DG) methods to approximate W2 , p strong solutions of second order linear elliptic partial differential equations (PDEs) in non-divergence form with continuous coefficients. The proposed IP-DG methods are closely related to the IP-DG methods for advection-diffusion equations, and they are easy to implement on existing standard IP-DG software platforms. It is proved that the proposed IP-DG methods have unique solutions and converge with optimal rate to the W2 , p strong solution in a discrete W2 , p-norm. The crux of the analysis is to establish a DG discrete counterpart of the Calderon–Zygmund estimate and to adapt a freezing coefficient technique used for the PDE analysis at the discrete level. To obtain such a crucial estimate, we need to establish broken W1 , p-norm error estimates for IP-DG approximations of constant coefficient elliptic PDEs, which is also of independent interest. Numerical experiments are provided to gauge the performance of the proposed IP-DG methods and to validate the theoretical convergence results.

Original languageEnglish
Pages (from-to)1651-1676
Number of pages26
JournalJournal of Scientific Computing
Volume74
Issue number3
DOIs
StatePublished - Mar 1 2018
Externally publishedYes

Funding

The work of the Xiaobing Feng and Stefan Schnake was partial supported by the NSF through Grant DMS-1318486 and the work of the Michael Neilan was partially supported by the NSF Grant DMS-1417980 and the Alfred Sloan Foundation.

Keywords

  • 35J25
  • 65N12
  • 65N30

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