Stabilized finite element method for the stationary mixed Stokes–Darcy problem
Abstract This paper considers numerical methods for solving the viscous incompressible steady-state Stokes–Darcy problem that can be implemented by the use of existing surface water and groundwater codes. In the porous medium problem for subsurface flow, a mixed discretization, which describes the m...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2018-09-01
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| Series: | Advances in Difference Equations |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13662-018-1809-2 |
| _version_ | 1819078577808736256 |
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| author | Jiaping Yu Md. Abdullah Al Mahbub Feng Shi Haibiao Zheng |
| author_facet | Jiaping Yu Md. Abdullah Al Mahbub Feng Shi Haibiao Zheng |
| author_sort | Jiaping Yu |
| collection | DOAJ |
| description | Abstract This paper considers numerical methods for solving the viscous incompressible steady-state Stokes–Darcy problem that can be implemented by the use of existing surface water and groundwater codes. In the porous medium problem for subsurface flow, a mixed discretization, which describes the macroscopic properties of a filtration process and is vigorous with respect to the variations in the material data, is often advocated. However, the theory of mixed spacial discretizations to Stokes–Darcy problems is far less developed than non-mixed versions. We develop herein a new robust stabilized fully mixed discretization technique in the porous media region coupled with the fluid region via the physically appropriate coupling conditions on the interface. The method developed here does not use any Lagrange multiplier and introduces a stabilization term in the temporal discretization to ensure the stability of the finite element scheme. The well-posedness of the finite element scheme and its convergence analysis are also derived. Finally, the efficiency and accuracy of the numerical methods are illustrated by several testing examples. |
| first_indexed | 2024-12-21T19:15:19Z |
| format | Article |
| id | doaj.art-6fd6863831c44bcdb7034696c8dffb19 |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-21T19:15:19Z |
| publishDate | 2018-09-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-6fd6863831c44bcdb7034696c8dffb192022-12-21T18:53:04ZengSpringerOpenAdvances in Difference Equations1687-18472018-09-012018111910.1186/s13662-018-1809-2Stabilized finite element method for the stationary mixed Stokes–Darcy problemJiaping Yu0Md. Abdullah Al Mahbub1Feng Shi2Haibiao Zheng3School of Science, Donghua UniversitySchool of Mathematical Sciences, East China Normal UniversityCollege of Science, Harbin Institute of TechnologySchool of Mathematical Sciences, East China Normal UniversityAbstract This paper considers numerical methods for solving the viscous incompressible steady-state Stokes–Darcy problem that can be implemented by the use of existing surface water and groundwater codes. In the porous medium problem for subsurface flow, a mixed discretization, which describes the macroscopic properties of a filtration process and is vigorous with respect to the variations in the material data, is often advocated. However, the theory of mixed spacial discretizations to Stokes–Darcy problems is far less developed than non-mixed versions. We develop herein a new robust stabilized fully mixed discretization technique in the porous media region coupled with the fluid region via the physically appropriate coupling conditions on the interface. The method developed here does not use any Lagrange multiplier and introduces a stabilization term in the temporal discretization to ensure the stability of the finite element scheme. The well-posedness of the finite element scheme and its convergence analysis are also derived. Finally, the efficiency and accuracy of the numerical methods are illustrated by several testing examples.http://link.springer.com/article/10.1186/s13662-018-1809-2Stokes–Darcy problemMixed finite elementFree flowPorous media flowStabilized scheme |
| spellingShingle | Jiaping Yu Md. Abdullah Al Mahbub Feng Shi Haibiao Zheng Stabilized finite element method for the stationary mixed Stokes–Darcy problem Advances in Difference Equations Stokes–Darcy problem Mixed finite element Free flow Porous media flow Stabilized scheme |
| title | Stabilized finite element method for the stationary mixed Stokes–Darcy problem |
| title_full | Stabilized finite element method for the stationary mixed Stokes–Darcy problem |
| title_fullStr | Stabilized finite element method for the stationary mixed Stokes–Darcy problem |
| title_full_unstemmed | Stabilized finite element method for the stationary mixed Stokes–Darcy problem |
| title_short | Stabilized finite element method for the stationary mixed Stokes–Darcy problem |
| title_sort | stabilized finite element method for the stationary mixed stokes darcy problem |
| topic | Stokes–Darcy problem Mixed finite element Free flow Porous media flow Stabilized scheme |
| url | http://link.springer.com/article/10.1186/s13662-018-1809-2 |
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