Interior Estimates of Semidiscrete Finite Element Methods for Parabolic Problems with Distributional Data

Li GUo Wayne State University Hengguang Li Wayne State University Yang Yang Michigan Technological University

Numerical Analysis and Scientific Computing mathscidoc:1907.25007

Journal of Computational Mathematics, 37, 460-476, 2019
Let $\Omega\subset\mathbb R^d$, $1\leq d\leq 3$, be a bounded $d$-polytope. Consider the parabolic equation on $\Omega$ with the Dirac delta function on the right hand side. We study the well-posedness, regularity, and the interior error estimate of semidiscrete finite element approximations of the equation. In particular, we derive that the interior error is bounded by the best local approximation error, the negative norms of the error, and the negative norms of the time derivative of the error. This result implies different convergence rates for the numerical solution in different interior regions, especially when the region is close to the singular point. Numerical test results are reported to support the theoretical prediction.
Parabolic problems, polyhedral domain, finite element methods, interior estimates, Sobolev norms, singularity
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@inproceedings{li2019interior,
  title={Interior Estimates of Semidiscrete Finite Element Methods for Parabolic Problems with Distributional Data},
  author={Li GUo, Hengguang Li, and Yang Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190701095610944186387},
  booktitle={Journal of Computational Mathematics},
  volume={37},
  pages={460-476},
  year={2019},
}
Li GUo, Hengguang Li, and Yang Yang. Interior Estimates of Semidiscrete Finite Element Methods for Parabolic Problems with Distributional Data. 2019. Vol. 37. In Journal of Computational Mathematics. pp.460-476. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190701095610944186387.
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