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Volume 17, Issue 6
An Absolutely Stabilized Virtual Element Method for the Incompressible Stokes Equations

Xi Zhang & Minfu Feng

Adv. Appl. Math. Mech., 17 (2025), pp. 1625-1653.

Published online: 2025-09

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  • Abstract

In this paper, we propose an absolutely stabilized mixed virtual element method (mixed VEM) for the incompressible Stokes equations. We employ the $C^0$-conforming virtual element space for the velocity and discontinuous polynomial space for the pressure. The velocity-pressure pair contains"arbitrary-order" polynomials, and stability is guaranteed by the"absolutely stabilized finite element formulation" originally introduced in [21]. We establish error estimates in the energy norm for both velocity and pressure, as well as an $L^2-$ error estimate for the velocity. Several numerical experiments validate the theoretical analysis.

  • AMS Subject Headings

65N30, 76D07, 35Q35

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-17-1625, author = {Zhang , Xi and Feng , Minfu}, title = {An Absolutely Stabilized Virtual Element Method for the Incompressible Stokes Equations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2025}, volume = {17}, number = {6}, pages = {1625--1653}, abstract = {

In this paper, we propose an absolutely stabilized mixed virtual element method (mixed VEM) for the incompressible Stokes equations. We employ the $C^0$-conforming virtual element space for the velocity and discontinuous polynomial space for the pressure. The velocity-pressure pair contains"arbitrary-order" polynomials, and stability is guaranteed by the"absolutely stabilized finite element formulation" originally introduced in [21]. We establish error estimates in the energy norm for both velocity and pressure, as well as an $L^2-$ error estimate for the velocity. Several numerical experiments validate the theoretical analysis.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2023-0281}, url = {http://global-sci.org/intro/article_detail/aamm/24489.html} }
TY - JOUR T1 - An Absolutely Stabilized Virtual Element Method for the Incompressible Stokes Equations AU - Zhang , Xi AU - Feng , Minfu JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 1625 EP - 1653 PY - 2025 DA - 2025/09 SN - 17 DO - http://doi.org/10.4208/aamm.OA-2023-0281 UR - https://global-sci.org/intro/article_detail/aamm/24489.html KW - Absolutely stabilized mixed VEM, incompressible Stokes equation, arbitrary-order polynomials. AB -

In this paper, we propose an absolutely stabilized mixed virtual element method (mixed VEM) for the incompressible Stokes equations. We employ the $C^0$-conforming virtual element space for the velocity and discontinuous polynomial space for the pressure. The velocity-pressure pair contains"arbitrary-order" polynomials, and stability is guaranteed by the"absolutely stabilized finite element formulation" originally introduced in [21]. We establish error estimates in the energy norm for both velocity and pressure, as well as an $L^2-$ error estimate for the velocity. Several numerical experiments validate the theoretical analysis.

Zhang , Xi and Feng , Minfu. (2025). An Absolutely Stabilized Virtual Element Method for the Incompressible Stokes Equations. Advances in Applied Mathematics and Mechanics. 17 (6). 1625-1653. doi:10.4208/aamm.OA-2023-0281
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