Uniform Finite Element Error Estimates with Power-Type Asymptotic Constants for Unsteady Navier–Stokes Equations
Uniform error estimates with power-type asymptotic constants of the finite element method for the unsteady Navier–Stokes equations are deduced in this paper. By introducing an iterative scheme and studying its convergence, we firstly derive that the solution of the Navier–Stokes equations is bounded...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-07-01
|
| Series: | Entropy |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1099-4300/24/7/948 |
| _version_ | 1797406517186527232 |
|---|---|
| author | Cong Xie Kun Wang |
| author_facet | Cong Xie Kun Wang |
| author_sort | Cong Xie |
| collection | DOAJ |
| description | Uniform error estimates with power-type asymptotic constants of the finite element method for the unsteady Navier–Stokes equations are deduced in this paper. By introducing an iterative scheme and studying its convergence, we firstly derive that the solution of the Navier–Stokes equations is bounded by power-type constants, where we avoid applying the Gronwall lemma, which generates exponential-type factors. Then, the technique is extended to the error estimate of the long-time finite element approximation. The analyses show that, under some assumptions on the given data, the asymptotic constants in the finite element error estimates for the unsteady Navier–Stokes equations are uniformly power functions with respect to the initial data, the viscosity, and the body force for all time <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. Finally, some numerical examples are shown to verify the theoretical predictions. |
| first_indexed | 2024-03-09T03:27:36Z |
| format | Article |
| id | doaj.art-a1f651f3da4d4f52a679106b79ebf34e |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-09T03:27:36Z |
| publishDate | 2022-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-a1f651f3da4d4f52a679106b79ebf34e2023-12-03T15:00:31ZengMDPI AGEntropy1099-43002022-07-0124794810.3390/e24070948Uniform Finite Element Error Estimates with Power-Type Asymptotic Constants for Unsteady Navier–Stokes EquationsCong Xie0Kun Wang1College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, ChinaCollege of Mathematics and Statistics, Chongqing University, Chongqing 401331, ChinaUniform error estimates with power-type asymptotic constants of the finite element method for the unsteady Navier–Stokes equations are deduced in this paper. By introducing an iterative scheme and studying its convergence, we firstly derive that the solution of the Navier–Stokes equations is bounded by power-type constants, where we avoid applying the Gronwall lemma, which generates exponential-type factors. Then, the technique is extended to the error estimate of the long-time finite element approximation. The analyses show that, under some assumptions on the given data, the asymptotic constants in the finite element error estimates for the unsteady Navier–Stokes equations are uniformly power functions with respect to the initial data, the viscosity, and the body force for all time <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. Finally, some numerical examples are shown to verify the theoretical predictions.https://www.mdpi.com/1099-4300/24/7/948Navier–Stokes equationspower-type asymptotic constantlong-time stabilityfinite element methoderror estimate |
| spellingShingle | Cong Xie Kun Wang Uniform Finite Element Error Estimates with Power-Type Asymptotic Constants for Unsteady Navier–Stokes Equations Entropy Navier–Stokes equations power-type asymptotic constant long-time stability finite element method error estimate |
| title | Uniform Finite Element Error Estimates with Power-Type Asymptotic Constants for Unsteady Navier–Stokes Equations |
| title_full | Uniform Finite Element Error Estimates with Power-Type Asymptotic Constants for Unsteady Navier–Stokes Equations |
| title_fullStr | Uniform Finite Element Error Estimates with Power-Type Asymptotic Constants for Unsteady Navier–Stokes Equations |
| title_full_unstemmed | Uniform Finite Element Error Estimates with Power-Type Asymptotic Constants for Unsteady Navier–Stokes Equations |
| title_short | Uniform Finite Element Error Estimates with Power-Type Asymptotic Constants for Unsteady Navier–Stokes Equations |
| title_sort | uniform finite element error estimates with power type asymptotic constants for unsteady navier stokes equations |
| topic | Navier–Stokes equations power-type asymptotic constant long-time stability finite element method error estimate |
| url | https://www.mdpi.com/1099-4300/24/7/948 |
| work_keys_str_mv | AT congxie uniformfiniteelementerrorestimateswithpowertypeasymptoticconstantsforunsteadynavierstokesequations AT kunwang uniformfiniteelementerrorestimateswithpowertypeasymptoticconstantsforunsteadynavierstokesequations |