A Fully-Mixed Finite Element Method for the n-Dimensional Boussinesq Problem with Temperature-Dependent Parameters

In this paper, we introduce and analyze a high-order, fully-mixed finite element method for the free convection of -dimensional fluids, , with temperature-dependent viscosity and thermal conductivity. The mathematical model is given by the coupling of the equations of continuity, momentum (Navier–St...

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Vydáno v:Journal of computational methods in applied mathematics Ročník 20; číslo 2; s. 187 - 213
Hlavní autoři: Almonacid, Javier A., Gatica, Gabriel N.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Minsk De Gruyter 01.04.2020
Walter de Gruyter GmbH
Témata:
35Q79
65N12
65N15
65N30
76D07
76R05
80A20
A Priori Error Analysis
Augmented Fully-Mixed Formulation
Boundary conditions
Boussinesq approximation
Boussinesq Equations
Computational fluid dynamics
Exact solutions
Finite element analysis
Finite element method
Finite Element Methods
Fixed points (mathematics)
Fixed-point Theory
Free convection
Galerkin method
Temperature dependence
Temperature gradients
Tensors
Thermal conductivity
Vectors (mathematics)
Vorticity
Well posed problems
ISSN:1609-4840, 1609-9389
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Abstract In this paper, we introduce and analyze a high-order, fully-mixed finite element method for the free convection of -dimensional fluids, , with temperature-dependent viscosity and thermal conductivity. The mathematical model is given by the coupling of the equations of continuity, momentum (Navier–Stokes) and energy by means of the Boussinesq approximation, as well as mixed thermal boundary conditions and a Dirichlet condition on the velocity. Because of the dependence on the temperature of the fluid properties, several additional variables are defined, thus resulting in an augmented formulation that seeks the rate of strain, pseudostress and vorticity tensors, velocity, temperature gradient and pseudoheat vectors, and temperature of the fluid. Using a fixed-point approach, smallness-of-data assumptions and a slight higher-regularity assumption for the exact solution provide the necessary well-posedness results at both continuous and discrete levels. In addition, and as a result of the augmentation, no discrete inf-sup conditions are needed for the well-posedness of the Galerkin scheme, which provides freedom of choice with respect to the finite element spaces. In particular, we suggest a combination based on Raviart–Thomas, Lagrange and discontinuous elements for which we derive optimal a priori error estimates. Finally, several numerical examples illustrating the performance of the method and confirming the theoretical rates of convergence are reported.
AbstractList In this paper, we introduce and analyze a high-order, fully-mixed finite element method for the free convection of n -dimensional fluids, n ∈ { 2 , 3 } {n\in\{2,3\}} , with temperature-dependent viscosity and thermal conductivity. The mathematical model is given by the coupling of the equations of continuity, momentum (Navier–Stokes) and energy by means of the Boussinesq approximation, as well as mixed thermal boundary conditions and a Dirichlet condition on the velocity. Because of the dependence on the temperature of the fluid properties, several additional variables are defined, thus resulting in an augmented formulation that seeks the rate of strain, pseudostress and vorticity tensors, velocity, temperature gradient and pseudoheat vectors, and temperature of the fluid. Using a fixed-point approach, smallness-of-data assumptions and a slight higher-regularity assumption for the exact solution provide the necessary well-posedness results at both continuous and discrete levels. In addition, and as a result of the augmentation, no discrete inf-sup conditions are needed for the well-posedness of the Galerkin scheme, which provides freedom of choice with respect to the finite element spaces. In particular, we suggest a combination based on Raviart–Thomas, Lagrange and discontinuous elements for which we derive optimal a priori error estimates. Finally, several numerical examples illustrating the performance of the method and confirming the theoretical rates of convergence are reported.
In this paper, we introduce and analyze a high-order, fully-mixed finite element method for the free convection ofn-dimensional fluids, n∈{2,3}{n\in\{2,3\}}, with temperature-dependent viscosity and thermal conductivity.The mathematical model is given by the coupling of the equations of continuity, momentum (Navier–Stokes)and energy by means of the Boussinesq approximation, as well as mixed thermal boundary conditions and aDirichlet condition on the velocity. Because of the dependence on the temperature ofthe fluid properties, several additional variables are defined, thus resulting in an augmented formulationthat seeks the rate of strain, pseudostress and vorticity tensors,velocity, temperature gradient and pseudoheat vectors, and temperature of the fluid. Using a fixed-point approach,smallness-of-data assumptions and a slight higher-regularity assumption for the exact solution provide the necessarywell-posedness results at both continuous and discrete levels. In addition, and as a result of the augmentation,no discrete inf-sup conditions are needed for the well-posedness of the Galerkin scheme, which provides freedom of choicewith respect to the finite element spaces. In particular, we suggest a combination based on Raviart–Thomas, Lagrange anddiscontinuous elements for which we derive optimal a priori error estimates. Finally, several numerical examplesillustrating the performance of the method and confirming the theoretical rates of convergence are reported.
In this paper, we introduce and analyze a high-order, fully-mixed finite element method for the free convection of -dimensional fluids, , with temperature-dependent viscosity and thermal conductivity. The mathematical model is given by the coupling of the equations of continuity, momentum (Navier–Stokes) and energy by means of the Boussinesq approximation, as well as mixed thermal boundary conditions and a Dirichlet condition on the velocity. Because of the dependence on the temperature of the fluid properties, several additional variables are defined, thus resulting in an augmented formulation that seeks the rate of strain, pseudostress and vorticity tensors, velocity, temperature gradient and pseudoheat vectors, and temperature of the fluid. Using a fixed-point approach, smallness-of-data assumptions and a slight higher-regularity assumption for the exact solution provide the necessary well-posedness results at both continuous and discrete levels. In addition, and as a result of the augmentation, no discrete inf-sup conditions are needed for the well-posedness of the Galerkin scheme, which provides freedom of choice with respect to the finite element spaces. In particular, we suggest a combination based on Raviart–Thomas, Lagrange and discontinuous elements for which we derive optimal a priori error estimates. Finally, several numerical examples illustrating the performance of the method and confirming the theoretical rates of convergence are reported.
Author Gatica, Gabriel N.
Almonacid, Javier A.
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Snippet In this paper, we introduce and analyze a high-order, fully-mixed finite element method for the free convection of -dimensional fluids, , with...
In this paper, we introduce and analyze a high-order, fully-mixed finite element method for the free convection of n -dimensional fluids, n ∈ { 2 , 3 }...
In this paper, we introduce and analyze a high-order, fully-mixed finite element method for the free convection ofn-dimensional fluids, n∈{2,3}{n\in\{2,3\}},...
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SubjectTerms 35Q79
65N12
65N15
65N30
76D07
76R05
80A20
A Priori Error Analysis
Augmented Fully-Mixed Formulation
Boundary conditions
Boussinesq approximation
Boussinesq Equations
Computational fluid dynamics
Exact solutions
Finite element analysis
Finite element method
Finite Element Methods
Fixed points (mathematics)
Fixed-point Theory
Free convection
Galerkin method
Temperature dependence
Temperature gradients
Tensors
Thermal conductivity
Vectors (mathematics)
Vorticity
Well posed problems
Title A Fully-Mixed Finite Element Method for the n-Dimensional Boussinesq Problem with Temperature-Dependent Parameters
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