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Volume 42, Issue 1
Unconditional Convergence and Error Estimates of a Fully Discrete Finite Element Method for the Micropolar Navier-Stokes Equations

Shipeng Mao, Jiaao Sun & Wendong Xue

DOI: 10.4208/jcm.2201-m2021-0315

J. Comp. Math., 42 (2024), pp. 71-110.

Published online: 2023-12

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  • Abstract

In this paper, we consider the initial-boundary value problem (IBVP) for the micropolar Navier-Stokes equations (MNSE) and analyze a first order fully discrete mixed finite element scheme. We first establish some regularity results for the solution of MNSE, which seem to be not available in the literature. Next, we study a semi-implicit time-discrete scheme for the MNSE and prove $\boldsymbol{L}^2-\boldsymbol{H}^1$ error estimates for the time discrete solution. Furthermore, certain regularity results for the time discrete solution are established rigorously. Based on these regularity results, we prove the unconditional $\boldsymbol{L}^2-\boldsymbol{H}^1$ error estimates for the finite element solution of MNSE. Finally, some numerical examples are carried out to demonstrate both accuracy and efficiency of the fully discrete finite element scheme.

  • Keywords

Micropolar fluids, Regularity estimates, Euler semi-implicit scheme, Mixed finite element methods, Unconditional convergence.

  • AMS Subject Headings

76M10, 65M12, 65M15, 65M60.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-71, author = {Mao , ShipengSun , Jiaao and Xue , Wendong}, title = {Unconditional Convergence and Error Estimates of a Fully Discrete Finite Element Method for the Micropolar Navier-Stokes Equations}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {42}, number = {1}, pages = {71--110}, abstract = {

In this paper, we consider the initial-boundary value problem (IBVP) for the micropolar Navier-Stokes equations (MNSE) and analyze a first order fully discrete mixed finite element scheme. We first establish some regularity results for the solution of MNSE, which seem to be not available in the literature. Next, we study a semi-implicit time-discrete scheme for the MNSE and prove $\boldsymbol{L}^2-\boldsymbol{H}^1$ error estimates for the time discrete solution. Furthermore, certain regularity results for the time discrete solution are established rigorously. Based on these regularity results, we prove the unconditional $\boldsymbol{L}^2-\boldsymbol{H}^1$ error estimates for the finite element solution of MNSE. Finally, some numerical examples are carried out to demonstrate both accuracy and efficiency of the fully discrete finite element scheme.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2201-m2021-0315}, url = {http://global-sci.org/intro/article_detail/jcm/22153.html} }
TY - JOUR T1 - Unconditional Convergence and Error Estimates of a Fully Discrete Finite Element Method for the Micropolar Navier-Stokes Equations AU - Mao , Shipeng AU - Sun , Jiaao AU - Xue , Wendong JO - Journal of Computational Mathematics VL - 1 SP - 71 EP - 110 PY - 2023 DA - 2023/12 SN - 42 DO - http://doi.org/10.4208/jcm.2201-m2021-0315 UR - https://global-sci.org/intro/article_detail/jcm/22153.html KW - Micropolar fluids, Regularity estimates, Euler semi-implicit scheme, Mixed finite element methods, Unconditional convergence. AB -

In this paper, we consider the initial-boundary value problem (IBVP) for the micropolar Navier-Stokes equations (MNSE) and analyze a first order fully discrete mixed finite element scheme. We first establish some regularity results for the solution of MNSE, which seem to be not available in the literature. Next, we study a semi-implicit time-discrete scheme for the MNSE and prove $\boldsymbol{L}^2-\boldsymbol{H}^1$ error estimates for the time discrete solution. Furthermore, certain regularity results for the time discrete solution are established rigorously. Based on these regularity results, we prove the unconditional $\boldsymbol{L}^2-\boldsymbol{H}^1$ error estimates for the finite element solution of MNSE. Finally, some numerical examples are carried out to demonstrate both accuracy and efficiency of the fully discrete finite element scheme.

Shipeng Mao, Jiaao Sun & Wendong Xue. (2023). Unconditional Convergence and Error Estimates of a Fully Discrete Finite Element Method for the Micropolar Navier-Stokes Equations. Journal of Computational Mathematics. 42 (1). 71-110. doi:10.4208/jcm.2201-m2021-0315
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