A combined finite element method for parabolic equations posted in domains with rough boundaries

Abstract

In this paper, we consider the solution of parabolic equations posted in domains with rough boundaries using the combined finite element procedure proposed in Xu et al. [A combined finite element method for elliptic problems posted in domains with rough boundaries, J. Comput. Appl. Math. 336 (2018), pp. 235–248] in space and backward-Euler scheme in time. We first prove $L^2$ error estimates for elliptic projection operator $R_{h,H}$ for both $\text{β}=1$ and $\text{β}=-1$ by the dual argument which depend on the elliptic auxiliary problem, and then energy error estimates of semi-discrete and fully discrete schemes which have convergence rates about $O\left(H\right)+O\left(h^s\right)$ and $O\left(H\right)+O\left(h^s\right)+O\left(\mathrm{\Delta }t\right)$, where s>0, respectively. Numerical results are provided for parabolic equations in domains with non-oscillating or oscillating boundaries to verify the theoretical findings.

Keywords:

• Combined finite element method
• multiscale problem
• parabolic equation
• rough boundary

2010 Mathematics Subject Classifications:

• 34E13
• 65N15
• 65N30

Acknowledgements

The authors would like to thank the referees for their careful reading and constructive comments that improved the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of the authors was partially supported by the PhD Scientific Research Foundation of Jiangxi Science and Technology Normal University [grant number 2018BSQD009], the National Natural Science Foundation of China [grant number 61802157], and the Scientific Research Fund of Jiangxi Provincial Education Department [grant number GJJ180626].