Stable and unstable cross-grid Pk Ql mixed finite elements for the Stokes problem
In this paper we develop and analyze a family of mixed finite element methods for the numerical solution of the Stokes problem in two space dimensions. In these schemes, the pressure is interpolated on a mesh of rectangular elements, while the velocity is approximated on a triangular mesh obtained b...
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paper:paper_03770427_v234_n5_p1404_Armentano2023-06-08T15:38:56Z Stable and unstable cross-grid Pk Ql mixed finite elements for the Stokes problem Armentano, Maria Gabriela Cross-grid Macroelement technique Mixed finite elements Stability analysis Stokes problem A-stability Finite Element Macro element Mixed finite element methods Mixed finite elements Numerical results Numerical solution Pressure modes Stability analysis Stokes problem Triangular meshes Two space dimensions Numerical methods Stability Finite element method In this paper we develop and analyze a family of mixed finite element methods for the numerical solution of the Stokes problem in two space dimensions. In these schemes, the pressure is interpolated on a mesh of rectangular elements, while the velocity is approximated on a triangular mesh obtained by dividing each rectangle into four triangles by its diagonals. Continuous interpolations of degrees k for the velocity and l for the pressure are considered, so the new finite elements are called cross-grid Pk Ql. A stability analysis of these approximations is provided, based on the macroelement technique of Stenberg. The lowest order P1 Q1 and P2 Q1 cases are analyzed in detail; in the first case, a global spurious pressure mode is shown to exist, so this element is unstable. In the second case, however, stability is rigorously proved. Numerical results obtained in these two cases are also presented, which confirm the existence of the spurious pressure mode for the P1 Q1 element and the stability of the P2 Q1 element. © 2010 Elsevier B.V. All rights reserved. Fil:Armentano, M.G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2010 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03770427_v234_n5_p1404_Armentano http://hdl.handle.net/20.500.12110/paper_03770427_v234_n5_p1404_Armentano |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Cross-grid Macroelement technique Mixed finite elements Stability analysis Stokes problem A-stability Finite Element Macro element Mixed finite element methods Mixed finite elements Numerical results Numerical solution Pressure modes Stability analysis Stokes problem Triangular meshes Two space dimensions Numerical methods Stability Finite element method |
| spellingShingle |
Cross-grid Macroelement technique Mixed finite elements Stability analysis Stokes problem A-stability Finite Element Macro element Mixed finite element methods Mixed finite elements Numerical results Numerical solution Pressure modes Stability analysis Stokes problem Triangular meshes Two space dimensions Numerical methods Stability Finite element method Armentano, Maria Gabriela Stable and unstable cross-grid Pk Ql mixed finite elements for the Stokes problem |
| topic_facet |
Cross-grid Macroelement technique Mixed finite elements Stability analysis Stokes problem A-stability Finite Element Macro element Mixed finite element methods Mixed finite elements Numerical results Numerical solution Pressure modes Stability analysis Stokes problem Triangular meshes Two space dimensions Numerical methods Stability Finite element method |
| description |
In this paper we develop and analyze a family of mixed finite element methods for the numerical solution of the Stokes problem in two space dimensions. In these schemes, the pressure is interpolated on a mesh of rectangular elements, while the velocity is approximated on a triangular mesh obtained by dividing each rectangle into four triangles by its diagonals. Continuous interpolations of degrees k for the velocity and l for the pressure are considered, so the new finite elements are called cross-grid Pk Ql. A stability analysis of these approximations is provided, based on the macroelement technique of Stenberg. The lowest order P1 Q1 and P2 Q1 cases are analyzed in detail; in the first case, a global spurious pressure mode is shown to exist, so this element is unstable. In the second case, however, stability is rigorously proved. Numerical results obtained in these two cases are also presented, which confirm the existence of the spurious pressure mode for the P1 Q1 element and the stability of the P2 Q1 element. © 2010 Elsevier B.V. All rights reserved. |
| author |
Armentano, Maria Gabriela |
| author_facet |
Armentano, Maria Gabriela |
| author_sort |
Armentano, Maria Gabriela |
| title |
Stable and unstable cross-grid Pk Ql mixed finite elements for the Stokes problem |
| title_short |
Stable and unstable cross-grid Pk Ql mixed finite elements for the Stokes problem |
| title_full |
Stable and unstable cross-grid Pk Ql mixed finite elements for the Stokes problem |
| title_fullStr |
Stable and unstable cross-grid Pk Ql mixed finite elements for the Stokes problem |
| title_full_unstemmed |
Stable and unstable cross-grid Pk Ql mixed finite elements for the Stokes problem |
| title_sort |
stable and unstable cross-grid pk ql mixed finite elements for the stokes problem |
| publishDate |
2010 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03770427_v234_n5_p1404_Armentano http://hdl.handle.net/20.500.12110/paper_03770427_v234_n5_p1404_Armentano |
| work_keys_str_mv |
AT armentanomariagabriela stableandunstablecrossgridpkqlmixedfiniteelementsforthestokesproblem |
| _version_ |
1768542461257318400 |