Volume 3, Issue 1
Adaptive $H$(div)-Conforming Embedded-Hybridized Discontinuous Galerkin Finite Element Methods for the Stokes Problems

Yihui Han & Haitao Leng

CSIAM Trans. Appl. Math., 3 (2022), pp. 82-108.

Published online: 2022-03

Export citation
  • Abstract

In this paper, we propose a residual-based a posteriori error estimator of embedded–hybridized discontinuous Galerkin finite element methods for the Stokes problems in two and three dimensions. The piecewise polynomials of degree $k (k≥1)$ and $k−1$ are used to approximate the velocity and pressure in the interior of elements, and the piecewise polynomials of degree $k$ are utilized to approximate the velocity and pressure on the inter-element boundaries. The attractive properties, named divergence-free and $H$(div)-conforming, are satisfied by the approximate velocity field. We prove that the a posteriori error estimator is robust in the sense that the ratio of the upper and lower bounds is independent of the mesh size and the viscosity. Finally, we provide several numerical examples to verify the theoretical results.

  • Keywords

Stokes equations, HDG methods, E-HDG methods, a posteriori error estimator, divergence-free, $H$(div)-conforming.

  • AMS Subject Headings

49M25, 65K10, 65M50

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{CSIAM-AM-3-82, author = {Yihui and Han and and 22463 and and Yihui Han and Haitao and Leng and and 22464 and and Haitao Leng}, title = {Adaptive $H$(div)-Conforming Embedded-Hybridized Discontinuous Galerkin Finite Element Methods for the Stokes Problems}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2022}, volume = {3}, number = {1}, pages = {82--108}, abstract = {

In this paper, we propose a residual-based a posteriori error estimator of embedded–hybridized discontinuous Galerkin finite element methods for the Stokes problems in two and three dimensions. The piecewise polynomials of degree $k (k≥1)$ and $k−1$ are used to approximate the velocity and pressure in the interior of elements, and the piecewise polynomials of degree $k$ are utilized to approximate the velocity and pressure on the inter-element boundaries. The attractive properties, named divergence-free and $H$(div)-conforming, are satisfied by the approximate velocity field. We prove that the a posteriori error estimator is robust in the sense that the ratio of the upper and lower bounds is independent of the mesh size and the viscosity. Finally, we provide several numerical examples to verify the theoretical results.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.SO-2021-0023}, url = {http://global-sci.org/intro/article_detail/csiam-am/20289.html} }
TY - JOUR T1 - Adaptive $H$(div)-Conforming Embedded-Hybridized Discontinuous Galerkin Finite Element Methods for the Stokes Problems AU - Han , Yihui AU - Leng , Haitao JO - CSIAM Transactions on Applied Mathematics VL - 1 SP - 82 EP - 108 PY - 2022 DA - 2022/03 SN - 3 DO - http://doi.org/10.4208/csiam-am.SO-2021-0023 UR - https://global-sci.org/intro/article_detail/csiam-am/20289.html KW - Stokes equations, HDG methods, E-HDG methods, a posteriori error estimator, divergence-free, $H$(div)-conforming. AB -

In this paper, we propose a residual-based a posteriori error estimator of embedded–hybridized discontinuous Galerkin finite element methods for the Stokes problems in two and three dimensions. The piecewise polynomials of degree $k (k≥1)$ and $k−1$ are used to approximate the velocity and pressure in the interior of elements, and the piecewise polynomials of degree $k$ are utilized to approximate the velocity and pressure on the inter-element boundaries. The attractive properties, named divergence-free and $H$(div)-conforming, are satisfied by the approximate velocity field. We prove that the a posteriori error estimator is robust in the sense that the ratio of the upper and lower bounds is independent of the mesh size and the viscosity. Finally, we provide several numerical examples to verify the theoretical results.

Yihui Han & Haitao Leng. (2022). Adaptive $H$(div)-Conforming Embedded-Hybridized Discontinuous Galerkin Finite Element Methods for the Stokes Problems. CSIAM Transactions on Applied Mathematics. 3 (1). 82-108. doi:10.4208/csiam-am.SO-2021-0023
Copy to clipboard
The citation has been copied to your clipboard