Finite Element Method for Time-Fractional Navier–Stokes Equations with Nonlinear Damping
We propose a hybrid numerical framework for solving time-fractional Navier–Stokes equations with nonlinear damping. The method combines the finite difference L1 scheme for time discretization of the Caputo derivative (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML"...
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-07-01
|
| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/9/7/445 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849406943981993984 |
|---|---|
| author | Shahid Hussain Xinlong Feng Arafat Hussain Ahmed Bakhet |
| author_facet | Shahid Hussain Xinlong Feng Arafat Hussain Ahmed Bakhet |
| author_sort | Shahid Hussain |
| collection | DOAJ |
| description | We propose a hybrid numerical framework for solving time-fractional Navier–Stokes equations with nonlinear damping. The method combines the finite difference L1 scheme for time discretization of the Caputo derivative (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>) with mixed finite element methods (P1b–P1 and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>2</mn></msub></semantics></math></inline-formula>–<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>1</mn></msub></semantics></math></inline-formula>) for spatial discretization of velocity and pressure. This approach addresses the key challenges of fractional models, including nonlocality and memory effects, while maintaining stability in the presence of the nonlinear damping term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>γ</mi><mo>|</mo><mi mathvariant="bold">u</mi><mo>|</mo></mrow><mrow><mi>r</mi><mo>−</mo><mn>2</mn></mrow></msup><mi mathvariant="bold">u</mi></mrow></semantics></math></inline-formula>, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>. We prove unconditional stability for both semi-discrete and fully discrete schemes and derive optimal error estimates for the velocity and pressure components. Numerical experiments validate the theoretical results. Convergence tests using exact solutions, along with benchmark problems such as backward-facing channel flow and lid-driven cavity flow, confirm the accuracy and reliability of the method. The computed velocity contours and streamlines show close agreement with analytical expectations. This scheme is particularly effective for capturing anomalous diffusion in Newtonian and turbulent flows, and it offers a strong foundation for future extensions to viscoelastic and biological fluid models. |
| format | Article |
| id | doaj-art-16fa91a59dba4b7692f89c7521175cd2 |
| institution | Kabale University |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-16fa91a59dba4b7692f89c7521175cd22025-08-20T03:36:14ZengMDPI AGFractal and Fractional2504-31102025-07-019744510.3390/fractalfract9070445Finite Element Method for Time-Fractional Navier–Stokes Equations with Nonlinear DampingShahid Hussain0Xinlong Feng1Arafat Hussain2Ahmed Bakhet3College of Mathematics and System Sciences, Xinjiang University, Urumqi 830049, ChinaCollege of Mathematics and System Sciences, Xinjiang University, Urumqi 830049, ChinaCollege of Mathematics and System Sciences, Xinjiang University, Urumqi 830049, ChinaCollege of Mathematics and System Sciences, Xinjiang University, Urumqi 830049, ChinaWe propose a hybrid numerical framework for solving time-fractional Navier–Stokes equations with nonlinear damping. The method combines the finite difference L1 scheme for time discretization of the Caputo derivative (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>) with mixed finite element methods (P1b–P1 and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>2</mn></msub></semantics></math></inline-formula>–<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>1</mn></msub></semantics></math></inline-formula>) for spatial discretization of velocity and pressure. This approach addresses the key challenges of fractional models, including nonlocality and memory effects, while maintaining stability in the presence of the nonlinear damping term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>γ</mi><mo>|</mo><mi mathvariant="bold">u</mi><mo>|</mo></mrow><mrow><mi>r</mi><mo>−</mo><mn>2</mn></mrow></msup><mi mathvariant="bold">u</mi></mrow></semantics></math></inline-formula>, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>. We prove unconditional stability for both semi-discrete and fully discrete schemes and derive optimal error estimates for the velocity and pressure components. Numerical experiments validate the theoretical results. Convergence tests using exact solutions, along with benchmark problems such as backward-facing channel flow and lid-driven cavity flow, confirm the accuracy and reliability of the method. The computed velocity contours and streamlines show close agreement with analytical expectations. This scheme is particularly effective for capturing anomalous diffusion in Newtonian and turbulent flows, and it offers a strong foundation for future extensions to viscoelastic and biological fluid models.https://www.mdpi.com/2504-3110/9/7/445time-fractional Navier–Stokes equationsCaputo fractional derivativesfinite element methodnonlinear dampingerror estimatesbackward flow |
| spellingShingle | Shahid Hussain Xinlong Feng Arafat Hussain Ahmed Bakhet Finite Element Method for Time-Fractional Navier–Stokes Equations with Nonlinear Damping Fractal and Fractional time-fractional Navier–Stokes equations Caputo fractional derivatives finite element method nonlinear damping error estimates backward flow |
| title | Finite Element Method for Time-Fractional Navier–Stokes Equations with Nonlinear Damping |
| title_full | Finite Element Method for Time-Fractional Navier–Stokes Equations with Nonlinear Damping |
| title_fullStr | Finite Element Method for Time-Fractional Navier–Stokes Equations with Nonlinear Damping |
| title_full_unstemmed | Finite Element Method for Time-Fractional Navier–Stokes Equations with Nonlinear Damping |
| title_short | Finite Element Method for Time-Fractional Navier–Stokes Equations with Nonlinear Damping |
| title_sort | finite element method for time fractional navier stokes equations with nonlinear damping |
| topic | time-fractional Navier–Stokes equations Caputo fractional derivatives finite element method nonlinear damping error estimates backward flow |
| url | https://www.mdpi.com/2504-3110/9/7/445 |
| work_keys_str_mv | AT shahidhussain finiteelementmethodfortimefractionalnavierstokesequationswithnonlineardamping AT xinlongfeng finiteelementmethodfortimefractionalnavierstokesequationswithnonlineardamping AT arafathussain finiteelementmethodfortimefractionalnavierstokesequationswithnonlineardamping AT ahmedbakhet finiteelementmethodfortimefractionalnavierstokesequationswithnonlineardamping |