Local and Parallel Stabilized Finite Element Methods Based on the Lowest Equal-Order Elements for the Stokes–Darcy Model
In this article, two kinds of local and parallel stabilized finite element methods based upon two grid discretizations are proposed and investigated for the Stokes–Darcy model. The lowest equal-order finite element pairs (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML&...
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| Format: | Article |
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MDPI AG
2023-11-01
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| Online Access: | https://www.mdpi.com/2227-7390/11/23/4820 |
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| author | Jing Han Guangzhi Du |
| author_facet | Jing Han Guangzhi Du |
| author_sort | Jing Han |
| collection | DOAJ |
| description | In this article, two kinds of local and parallel stabilized finite element methods based upon two grid discretizations are proposed and investigated for the Stokes–Darcy model. The lowest equal-order finite element pairs (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">P</mi><mn>1</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>-<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>) are taken into account to approximate the velocity, pressure, and piezometric head, respectively. To circumvent the inf-sup condition, the stabilized term is chosen as the difference between a consistent and an under-integrated mass matrix. The proposed algorithms consist of approximating the low-frequency component on the global coarse grid and the high-frequency component on the local fine grid and assembling them to obtain the final approximation. To obtain a global continuous solution, the technique tool of the partition of unity is used. A rigorous theoretical analysis for the algorithms was conducted and numerical experiments were carried out to indicate the validity and efficiency of the algorithms. |
| first_indexed | 2024-03-09T01:46:47Z |
| format | Article |
| id | doaj.art-4093f319e92c4bbba36bf5583d9575dd |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T01:46:47Z |
| publishDate | 2023-11-01 |
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| spelling | doaj.art-4093f319e92c4bbba36bf5583d9575dd2023-12-08T15:21:51ZengMDPI AGMathematics2227-73902023-11-011123482010.3390/math11234820Local and Parallel Stabilized Finite Element Methods Based on the Lowest Equal-Order Elements for the Stokes–Darcy ModelJing Han0Guangzhi Du1School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, ChinaSchool of Mathematics and Statistics, Shandong Normal University, Jinan 250014, ChinaIn this article, two kinds of local and parallel stabilized finite element methods based upon two grid discretizations are proposed and investigated for the Stokes–Darcy model. The lowest equal-order finite element pairs (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">P</mi><mn>1</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>-<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>) are taken into account to approximate the velocity, pressure, and piezometric head, respectively. To circumvent the inf-sup condition, the stabilized term is chosen as the difference between a consistent and an under-integrated mass matrix. The proposed algorithms consist of approximating the low-frequency component on the global coarse grid and the high-frequency component on the local fine grid and assembling them to obtain the final approximation. To obtain a global continuous solution, the technique tool of the partition of unity is used. A rigorous theoretical analysis for the algorithms was conducted and numerical experiments were carried out to indicate the validity and efficiency of the algorithms.https://www.mdpi.com/2227-7390/11/23/4820Stokes–Darcy modelstabilized finite element methodlocal and parallel finite element methodpartition of unity |
| spellingShingle | Jing Han Guangzhi Du Local and Parallel Stabilized Finite Element Methods Based on the Lowest Equal-Order Elements for the Stokes–Darcy Model Mathematics Stokes–Darcy model stabilized finite element method local and parallel finite element method partition of unity |
| title | Local and Parallel Stabilized Finite Element Methods Based on the Lowest Equal-Order Elements for the Stokes–Darcy Model |
| title_full | Local and Parallel Stabilized Finite Element Methods Based on the Lowest Equal-Order Elements for the Stokes–Darcy Model |
| title_fullStr | Local and Parallel Stabilized Finite Element Methods Based on the Lowest Equal-Order Elements for the Stokes–Darcy Model |
| title_full_unstemmed | Local and Parallel Stabilized Finite Element Methods Based on the Lowest Equal-Order Elements for the Stokes–Darcy Model |
| title_short | Local and Parallel Stabilized Finite Element Methods Based on the Lowest Equal-Order Elements for the Stokes–Darcy Model |
| title_sort | local and parallel stabilized finite element methods based on the lowest equal order elements for the stokes darcy model |
| topic | Stokes–Darcy model stabilized finite element method local and parallel finite element method partition of unity |
| url | https://www.mdpi.com/2227-7390/11/23/4820 |
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