A Wavelet-Based Adaptive Finite Element Method for the Stokes Problems
In this work, we present the mathematical formulation of the new adaptive multiresolution method for the Stokes problems of highly viscous materials arising in computational geodynamics. The method is based on particle-in-cell approach—the Stokes system is solved on a static Eulerian finite element...
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MDPI AG
2022-06-01
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| Series: | Fluids |
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| Online Access: | https://www.mdpi.com/2311-5521/7/7/221 |
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| author | Yury A. Mishin Oleg V. Vasilyev Taras V. Gerya |
| author_facet | Yury A. Mishin Oleg V. Vasilyev Taras V. Gerya |
| author_sort | Yury A. Mishin |
| collection | DOAJ |
| description | In this work, we present the mathematical formulation of the new adaptive multiresolution method for the Stokes problems of highly viscous materials arising in computational geodynamics. The method is based on particle-in-cell approach—the Stokes system is solved on a static Eulerian finite element grid and material properties are carried in space by Lagrangian material points. The Eulerian grid is adapted using the wavelet-based adaptation algorithm. Both bilinear (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="normal">Q</mi><mn>1</mn></msub><msub><mi mathvariant="normal">P</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="normal">Q</mi><mn>1</mn></msub><msub><mi mathvariant="normal">Q</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula>) and biquadratic (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="normal">Q</mi><mn>2</mn></msub><msub><mi mathvariant="normal">P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula>) mixed approximations for the Stokes system are supported. The proposed method is illustrated for a number of linear and nonlinear two-dimensional benchmark problems of geophysical relevance. The results of the adaptive numerical simulations using the proposed method are in an excellent agreement with those obtained on non-adaptive grids and with analytical solutions, while computational requirements are few orders of magnitude less compared to the non-adaptive simulations in terms of both time and memory usage. |
| first_indexed | 2024-03-09T03:26:49Z |
| format | Article |
| id | doaj.art-d99b33b21aa1408f9d910d65329b74b1 |
| institution | Directory Open Access Journal |
| issn | 2311-5521 |
| language | English |
| last_indexed | 2024-03-09T03:26:49Z |
| publishDate | 2022-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fluids |
| spelling | doaj.art-d99b33b21aa1408f9d910d65329b74b12023-12-03T15:02:05ZengMDPI AGFluids2311-55212022-06-017722110.3390/fluids7070221A Wavelet-Based Adaptive Finite Element Method for the Stokes ProblemsYury A. Mishin0Oleg V. Vasilyev1Taras V. Gerya2Department of Earth Sciences, Swiss Federal Institute of Technology Zürich (ETH Zürich), CH-8092 Zürich, SwitzerlandKeldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow 125047, RussiaDepartment of Earth Sciences, Swiss Federal Institute of Technology Zürich (ETH Zürich), CH-8092 Zürich, SwitzerlandIn this work, we present the mathematical formulation of the new adaptive multiresolution method for the Stokes problems of highly viscous materials arising in computational geodynamics. The method is based on particle-in-cell approach—the Stokes system is solved on a static Eulerian finite element grid and material properties are carried in space by Lagrangian material points. The Eulerian grid is adapted using the wavelet-based adaptation algorithm. Both bilinear (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="normal">Q</mi><mn>1</mn></msub><msub><mi mathvariant="normal">P</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="normal">Q</mi><mn>1</mn></msub><msub><mi mathvariant="normal">Q</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula>) and biquadratic (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="normal">Q</mi><mn>2</mn></msub><msub><mi mathvariant="normal">P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula>) mixed approximations for the Stokes system are supported. The proposed method is illustrated for a number of linear and nonlinear two-dimensional benchmark problems of geophysical relevance. The results of the adaptive numerical simulations using the proposed method are in an excellent agreement with those obtained on non-adaptive grids and with analytical solutions, while computational requirements are few orders of magnitude less compared to the non-adaptive simulations in terms of both time and memory usage.https://www.mdpi.com/2311-5521/7/7/221Stokes problemadaptive mesh refinementfinite element methodwaveletsparticle-in-cell |
| spellingShingle | Yury A. Mishin Oleg V. Vasilyev Taras V. Gerya A Wavelet-Based Adaptive Finite Element Method for the Stokes Problems Fluids Stokes problem adaptive mesh refinement finite element method wavelets particle-in-cell |
| title | A Wavelet-Based Adaptive Finite Element Method for the Stokes Problems |
| title_full | A Wavelet-Based Adaptive Finite Element Method for the Stokes Problems |
| title_fullStr | A Wavelet-Based Adaptive Finite Element Method for the Stokes Problems |
| title_full_unstemmed | A Wavelet-Based Adaptive Finite Element Method for the Stokes Problems |
| title_short | A Wavelet-Based Adaptive Finite Element Method for the Stokes Problems |
| title_sort | wavelet based adaptive finite element method for the stokes problems |
| topic | Stokes problem adaptive mesh refinement finite element method wavelets particle-in-cell |
| url | https://www.mdpi.com/2311-5521/7/7/221 |
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