Existence and Uniqueness of Generalized and Mixed Finite Element Solutions for Steady Boussinesq Equation
Herein, we mainly employ the fixed point theorem and Lax-Milgram theorem in functional analysis to prove the existence and uniqueness of generalized and mixed finite element (MFE) solutions for two-dimensional steady Boussinesq equation. Thus, we can fill in the gap of research for the steady Boussi...
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| Published in | Mathematics (Basel) Vol. 11; no. 3; p. 545 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
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Basel
MDPI AG
01.01.2023
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| ISSN | 2227-7390 2227-7390 |
| DOI | 10.3390/math11030545 |
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| Abstract | Herein, we mainly employ the fixed point theorem and Lax-Milgram theorem in functional analysis to prove the existence and uniqueness of generalized and mixed finite element (MFE) solutions for two-dimensional steady Boussinesq equation. Thus, we can fill in the gap of research for the steady Boussinesq equation since the existing studies for the equation are assumed the existence and uniqueness of generalized solution without providing proof. |
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| AbstractList | Herein, we mainly employ the fixed point theorem and Lax-Milgram theorem in functional analysis to prove the existence and uniqueness of generalized and mixed finite element (MFE) solutions for two-dimensional steady Boussinesq equation. Thus, we can fill in the gap of research for the steady Boussinesq equation since the existing studies for the equation are assumed the existence and uniqueness of generalized solution without providing proof. |
| Audience | Academic |
| Author | Luo, Zhendong Liu, Xiangdong Zeng, Yihui Li, Yuejie |
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| References | Hu (ref_3) 2016; 95 Lee (ref_10) 2000; 242 France (ref_1) 1989; 69 Alekseev (ref_5) 2011; 2011 Lee (ref_7) 2003; 19 ref_14 Huang (ref_4) 2014; 38 Baranovskii (ref_8) 2021; 189 ref_13 ref_12 ref_11 Colmenares (ref_6) 2017; 54 ref_2 ref_18 ref_17 Girault (ref_16) 1979; 33 ref_15 Baranovskii (ref_9) 2022; 86 |
| References_xml | – volume: 69 start-page: 89 year: 1989 ident: ref_1 article-title: Two classes of mixed finite element methods publication-title: Comput. Math. Appl. Mech. Eng. doi: 10.1016/0045-7825(88)90168-5 – volume: 95 start-page: 815 year: 2016 ident: ref_3 article-title: New Fourier-series-based analytical solution to the conduction-convection equation to calculate soil temperature, determine soil thermal properties, or estimate water flux publication-title: Int. J. Heat Mass Transf. doi: 10.1016/j.ijheatmasstransfer.2015.11.078 – volume: 242 start-page: 191 year: 2000 ident: ref_10 article-title: Analysis of optimal control problems for the 2-D stationary Boussinesq equations publication-title: J. Math. Anal. Appl. doi: 10.1006/jmaa.1999.6651 – ident: ref_2 – ident: ref_12 – volume: 33 start-page: 235 year: 1979 ident: ref_16 article-title: An analysis of a mixed finite element method for the Navier-Stokes equations publication-title: Numer. Math. doi: 10.1007/BF01398643 – volume: 54 start-page: 167 year: 2017 ident: ref_6 article-title: An augmented fully-mixed finite element method for the stationary Boussinesq problem publication-title: Calcolo doi: 10.1007/s10092-016-0182-3 – ident: ref_15 – volume: 2011 start-page: 535736 year: 2011 ident: ref_5 article-title: Stability of optimal controls for the stationary Boussinesq equations publication-title: Int. J. Differ. Equat. doi: 10.1155/2011/535736 – ident: ref_11 doi: 10.3390/fluids7060203 – ident: ref_13 – ident: ref_14 – ident: ref_17 – volume: 189 start-page: 623 year: 2021 ident: ref_8 article-title: Optimal boundary control of the Boussinesq approximation for polymeric fluids publication-title: J. Optim. Theory Appl. doi: 10.1007/s10957-021-01849-4 – ident: ref_18 – volume: 19 start-page: 255 year: 2003 ident: ref_7 article-title: Analysis and computational methods of Dirichlet boundary optimal control problems for 2D Boussinesq equations publication-title: Adv. Comput. Math. doi: 10.1023/A:1022872602498 – volume: 86 start-page: 221 year: 2022 ident: ref_9 article-title: The optimal start control problem for two-dimensional Boussinesq equations publication-title: Izv. Math. doi: 10.1070/IM9099 – volume: 38 start-page: 535 year: 2014 ident: ref_4 article-title: An Oseen scheme for the conduction-convection equations based on a stabilized nonconforming method publication-title: Appl. Math. Model. doi: 10.1016/j.apm.2013.06.033 |
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| SubjectTerms | Boussinesq equations existence and uniqueness of generalized solution existence and uniqueness of mixed finite element solution Existence theorems Finite element method Fixed point theory Fixed points (mathematics) Functional analysis Inequality Mathematical research Navier-Stokes equations Numerical analysis Reynolds number the fixed point theorem the Lax-Milgram theorem Theorems Uniqueness Water waves Wave equation |
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| Title | Existence and Uniqueness of Generalized and Mixed Finite Element Solutions for Steady Boussinesq Equation |
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