Finite Element Iterative Methods for the 3D Steady Navier--Stokes Equations
In this work, a finite element (FE) method is discussed for the 3D steady Navier–Stokes equations by using the finite element pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X<...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-12-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/23/12/1659 |
| _version_ | 1797504889401638912 |
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| author | Yinnian He |
| author_facet | Yinnian He |
| author_sort | Yinnian He |
| collection | DOAJ |
| description | In this work, a finite element (FE) method is discussed for the 3D steady Navier–Stokes equations by using the finite element pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>h</mi></msub><mo>×</mo><msub><mi>M</mi><mi>h</mi></msub></mrow></semantics></math></inline-formula>. The method consists of transmitting the finite element solution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>u</mi><mi>h</mi></msub><mo>,</mo><msub><mi>p</mi><mi>h</mi></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> of the 3D steady Navier–Stokes equations into the finite element solution pairs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>u</mi><mi>h</mi><mi>n</mi></msubsup><mo>,</mo><msubsup><mi>p</mi><mi>h</mi><mi>n</mi></msubsup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> based on the finite element space pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>h</mi></msub><mo>×</mo><msub><mi>M</mi><mi>h</mi></msub></mrow></semantics></math></inline-formula> of the 3D steady linearized Navier–Stokes equations by using the Stokes, Newton and Oseen iterative methods, where the finite element space pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>h</mi></msub><mo>×</mo><msub><mi>M</mi><mi>h</mi></msub></mrow></semantics></math></inline-formula> satisfies the discrete inf-sup condition in a 3D domain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula>. Here, we present the weak formulations of the FE method for solving the 3D steady Stokes, Newton and Oseen iterative equations, provide the existence and uniqueness of the FE solution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>u</mi><mi>h</mi><mi>n</mi></msubsup><mo>,</mo><msubsup><mi>p</mi><mi>h</mi><mi>n</mi></msubsup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> of the 3D steady Stokes, Newton and Oseen iterative equations, and deduce the convergence with respect to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>σ</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> of the FE solution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>u</mi><mi>h</mi><mi>n</mi></msubsup><mo>,</mo><msubsup><mi>p</mi><mi>h</mi><mi>n</mi></msubsup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> to the exact solution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> of the 3D steady Navier–Stokes equations in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mn>1</mn></msup><mo>−</mo><msup><mi>L</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula> norm. Finally, we also give the convergence order with respect to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>σ</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> of the FE velocity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>u</mi><mi>h</mi><mi>n</mi></msubsup></semantics></math></inline-formula> to the exact velocity <i>u</i> of the 3D steady Navier–Stokes equations in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> norm. |
| first_indexed | 2024-03-10T04:10:49Z |
| format | Article |
| id | doaj.art-18f680dafbe34ce597af1d0df741a9e9 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T04:10:49Z |
| publishDate | 2021-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-18f680dafbe34ce597af1d0df741a9e92023-11-23T08:11:17ZengMDPI AGEntropy1099-43002021-12-012312165910.3390/e23121659Finite Element Iterative Methods for the 3D Steady Navier--Stokes EquationsYinnian He0School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, ChinaIn this work, a finite element (FE) method is discussed for the 3D steady Navier–Stokes equations by using the finite element pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>h</mi></msub><mo>×</mo><msub><mi>M</mi><mi>h</mi></msub></mrow></semantics></math></inline-formula>. The method consists of transmitting the finite element solution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>u</mi><mi>h</mi></msub><mo>,</mo><msub><mi>p</mi><mi>h</mi></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> of the 3D steady Navier–Stokes equations into the finite element solution pairs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>u</mi><mi>h</mi><mi>n</mi></msubsup><mo>,</mo><msubsup><mi>p</mi><mi>h</mi><mi>n</mi></msubsup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> based on the finite element space pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>h</mi></msub><mo>×</mo><msub><mi>M</mi><mi>h</mi></msub></mrow></semantics></math></inline-formula> of the 3D steady linearized Navier–Stokes equations by using the Stokes, Newton and Oseen iterative methods, where the finite element space pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>h</mi></msub><mo>×</mo><msub><mi>M</mi><mi>h</mi></msub></mrow></semantics></math></inline-formula> satisfies the discrete inf-sup condition in a 3D domain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula>. Here, we present the weak formulations of the FE method for solving the 3D steady Stokes, Newton and Oseen iterative equations, provide the existence and uniqueness of the FE solution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>u</mi><mi>h</mi><mi>n</mi></msubsup><mo>,</mo><msubsup><mi>p</mi><mi>h</mi><mi>n</mi></msubsup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> of the 3D steady Stokes, Newton and Oseen iterative equations, and deduce the convergence with respect to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>σ</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> of the FE solution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>u</mi><mi>h</mi><mi>n</mi></msubsup><mo>,</mo><msubsup><mi>p</mi><mi>h</mi><mi>n</mi></msubsup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> to the exact solution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> of the 3D steady Navier–Stokes equations in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mn>1</mn></msup><mo>−</mo><msup><mi>L</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula> norm. Finally, we also give the convergence order with respect to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>σ</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> of the FE velocity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>u</mi><mi>h</mi><mi>n</mi></msubsup></semantics></math></inline-formula> to the exact velocity <i>u</i> of the 3D steady Navier–Stokes equations in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> norm.https://www.mdpi.com/1099-4300/23/12/1659Navier–Stokes equationsOseen iterative equationsNewton iterative equationsStokes iterative equationsweak formulationfinite element |
| spellingShingle | Yinnian He Finite Element Iterative Methods for the 3D Steady Navier--Stokes Equations Entropy Navier–Stokes equations Oseen iterative equations Newton iterative equations Stokes iterative equations weak formulation finite element |
| title | Finite Element Iterative Methods for the 3D Steady Navier--Stokes Equations |
| title_full | Finite Element Iterative Methods for the 3D Steady Navier--Stokes Equations |
| title_fullStr | Finite Element Iterative Methods for the 3D Steady Navier--Stokes Equations |
| title_full_unstemmed | Finite Element Iterative Methods for the 3D Steady Navier--Stokes Equations |
| title_short | Finite Element Iterative Methods for the 3D Steady Navier--Stokes Equations |
| title_sort | finite element iterative methods for the 3d steady navier stokes equations |
| topic | Navier–Stokes equations Oseen iterative equations Newton iterative equations Stokes iterative equations weak formulation finite element |
| url | https://www.mdpi.com/1099-4300/23/12/1659 |
| work_keys_str_mv | AT yinnianhe finiteelementiterativemethodsforthe3dsteadynavierstokesequations |