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Stability analysis and error estimates of local discontinuous Galerkin methods for convection–diffusion equations on overlapping meshes

  • Jie Du
  • Yang YangEmail author
  • Eric Chung
Article
  • 18 Downloads

Abstract

Local discontinuous Galerkin (LDG) methods are popular for convection–diffusion equations. In LDG methods, we introduce an auxiliary variable p to represent the derivative of the primary variable u, and solve them on the same mesh. In this paper, we will introduce a new LDG method, and solve u and p on different meshes. The stability and error estimates will be investigated. The new algorithm is more flexible and flux-free for pure diffusion equations without introducing additional computational cost compared with the original LDG methods, since it is not necessary to solve each equation twice. Moreover, it is possible to construct third-order maximum-principle-preserving schemes based on the new algorithm. However, one cannot anticipate optimal accuracy in some special cases. In this paper, we will find out the reason for accuracy degeneration which further leads to several alternatives to obtain optimal convergence rates. Finally, several numerical experiments will be given to demonstrate the stability and optimal accuracy of the new algorithm.

Keywords

Stability Error estimates Convection–diffusion equations Local discontinuous Galerkin method Overlapping meshes 

Mathematics Subject Classification

65M60 65M12 65M20 

Notes

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina
  2. 2.Department of Mathematical SciencesMichigan Technological UniversityHoughtonUSA
  3. 3.Department of MathematicsThe Chinese University of Hong KongSha TinHong Kong SAR

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