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

Jie Du Yau Mathematical Sciences Center, Tsinghua University Yang Yang Michigan Technological University Eric Chung Chinese University of Hongkong

Numerical Analysis and Scientific Computing mathscidoc:1907.25010

BIT Numerical Mathematics, 2019
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.
Stability, Error estimates, Convection-diffusion equations, Local discontinuous Galerkin method, Overlapping meshes.
[ Download ] [ 2019-07-01 10:05:35 uploaded by 0yyyzu ] [ 590 downloads ] [ 0 comments ]
  • https://link.springer.com/article/10.1007/s10543-019-00757-4
@inproceedings{jie2019stability,
  title={Stability analysis and error estimates of local discontinuous Galerkin methods for convection-diffusion equations on overlapping meshes},
  author={Jie Du, Yang Yang, and Eric Chung},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190701100536004514390},
  booktitle={BIT Numerical Mathematics},
  year={2019},
}
Jie Du, Yang Yang, and Eric Chung. Stability analysis and error estimates of local discontinuous Galerkin methods for convection-diffusion equations on overlapping meshes. 2019. In BIT Numerical Mathematics. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190701100536004514390.
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