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Volume 36, Issue 1
The Lowest-Order Stabilized Virtual Element Method for the Stokes Problem

Xin Liu, Qixuan Song, Yu Gao & Zhangxin Chen

Commun. Comput. Phys., 36 (2024), pp. 221-247.

Published online: 2024-07

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  • Abstract

In this paper, we develop and analyze two stabilized mixed virtual element schemes for the Stokes problem based on the lowest-order velocity-pressure pairs (i.e., a piecewise constant approximation for pressure and an approximation with an accuracy order $k = 1$ for velocity). By applying local pressure jump and projection stabilization, we ensure the well-posedness of our discrete schemes and obtain the corresponding optimal $H^1$- and $L^2$-error estimates. The proposed schemes offer a number of attractive computational properties, such as, the use of polygonal/polyhedral meshes (including non-convex and degenerate elements), yielding a symmetric linear system that involves neither the calculations of higher-order derivatives nor additional coupling terms, and being parameter-free in the local pressure projection stabilization. Finally, we present the matrix implementations of the essential ingredients of our stabilized virtual element methods and investigate two- and three-dimensional numerical experiments for incompressible flow to show the performance of these numerical schemes.

  • AMS Subject Headings

65N30, 65N12, 65N15, 76D05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-36-221, author = {Liu , XinSong , QixuanGao , Yu and Chen , Zhangxin}, title = {The Lowest-Order Stabilized Virtual Element Method for the Stokes Problem}, journal = {Communications in Computational Physics}, year = {2024}, volume = {36}, number = {1}, pages = {221--247}, abstract = {

In this paper, we develop and analyze two stabilized mixed virtual element schemes for the Stokes problem based on the lowest-order velocity-pressure pairs (i.e., a piecewise constant approximation for pressure and an approximation with an accuracy order $k = 1$ for velocity). By applying local pressure jump and projection stabilization, we ensure the well-posedness of our discrete schemes and obtain the corresponding optimal $H^1$- and $L^2$-error estimates. The proposed schemes offer a number of attractive computational properties, such as, the use of polygonal/polyhedral meshes (including non-convex and degenerate elements), yielding a symmetric linear system that involves neither the calculations of higher-order derivatives nor additional coupling terms, and being parameter-free in the local pressure projection stabilization. Finally, we present the matrix implementations of the essential ingredients of our stabilized virtual element methods and investigate two- and three-dimensional numerical experiments for incompressible flow to show the performance of these numerical schemes.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0233}, url = {http://global-sci.org/intro/article_detail/cicp/23302.html} }
TY - JOUR T1 - The Lowest-Order Stabilized Virtual Element Method for the Stokes Problem AU - Liu , Xin AU - Song , Qixuan AU - Gao , Yu AU - Chen , Zhangxin JO - Communications in Computational Physics VL - 1 SP - 221 EP - 247 PY - 2024 DA - 2024/07 SN - 36 DO - http://doi.org/10.4208/cicp.OA-2023-0233 UR - https://global-sci.org/intro/article_detail/cicp/23302.html KW - Stokes equations, stabilized virtual element scheme, pressure jump, pressure projection, polygonal meshes. AB -

In this paper, we develop and analyze two stabilized mixed virtual element schemes for the Stokes problem based on the lowest-order velocity-pressure pairs (i.e., a piecewise constant approximation for pressure and an approximation with an accuracy order $k = 1$ for velocity). By applying local pressure jump and projection stabilization, we ensure the well-posedness of our discrete schemes and obtain the corresponding optimal $H^1$- and $L^2$-error estimates. The proposed schemes offer a number of attractive computational properties, such as, the use of polygonal/polyhedral meshes (including non-convex and degenerate elements), yielding a symmetric linear system that involves neither the calculations of higher-order derivatives nor additional coupling terms, and being parameter-free in the local pressure projection stabilization. Finally, we present the matrix implementations of the essential ingredients of our stabilized virtual element methods and investigate two- and three-dimensional numerical experiments for incompressible flow to show the performance of these numerical schemes.

Xin Liu, Qixuan Song, Yu Gao & Zhangxin Chen. (2024). The Lowest-Order Stabilized Virtual Element Method for the Stokes Problem. Communications in Computational Physics. 36 (1). 221-247. doi:10.4208/cicp.OA-2023-0233
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