Superconvergence of the local discontinuous Galerkin method for linear fourth-order time-dependent problems in one space dimension

Xiong Meng Harbin Institute of Technology Chi-Wang Shu Brown University Boying Wu Harbin Institute of Technology

Numerical Analysis and Scientific Computing mathscidoc:1610.25081

IMA Journal of Numerical Analysis, 32, 1294-1328, 2012
In this paper, we investigate the superconvergence property of the local discontinuous Galerkin (LDG) methods for solving one-dimensional linear time dependent fourth order problems. We prove that the error between the LDG solution and a particular projection of the exact solution, $\bar{e}_u$, achieves $(k+\frac32)$-th order superconvergence when polynomials of degree $k$ ($k\ge 1$) are used. Numerical experiments of $P^k$ polynomials, with $1\le k \le3$, are displayed to demonstrate the theoretical results, which show that the error $\bar{e}_u$ actually achieves $(k+2)$-th order superconvergence, indicating that the error bound for $\bar{e}_u$ obtained in this paper is sub-optimal. Initial-boundary value problems, nonlinear equations and solutions having singularities are numerically investigated to verify that the conclusions hold true for very general cases.
local discontinuous Galerkin method, superconvergence, fourth order problems, error estimates
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@inproceedings{xiong2012superconvergence,
  title={Superconvergence of the local discontinuous Galerkin method for linear fourth-order time-dependent problems in one space dimension},
  author={Xiong Meng, Chi-Wang Shu, and Boying Wu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161012114817050801199},
  booktitle={IMA Journal of Numerical Analysis},
  volume={32},
  pages={1294-1328},
  year={2012},
}
Xiong Meng, Chi-Wang Shu, and Boying Wu. Superconvergence of the local discontinuous Galerkin method for linear fourth-order time-dependent problems in one space dimension. 2012. Vol. 32. In IMA Journal of Numerical Analysis. pp.1294-1328. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161012114817050801199.
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