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A fully-mixed finite element method for the steady state Oberbeck-Boussinesq system. (English) Zbl 07219867
The rough contents of this vast paper are as follows: 1. Introduction, 2. Governing equations, 3. Well-posedness of the continuous problem, 4. The Galerkin scheme, 5. A priori error analysis, 6. A posteriori error estimator and 7. Numerical results. The References contain 48 entries. The authors are concerned with the construction, the analysis and the Implementation of a high-order optimally convergent mixed finite element scheme for solving various nonlinearly coupled system from fluid dynamics. Thus, the stress and the velocity gradient in the fluid equations, the temperature gradient, the concentration gradient and a vector version of the Bernoulli tensor combining advective and diffusive heat and concentration fluxes are the field variables (unknowns). From mathematical point of view the well-known Banach and Brouwer theorems, combined with the application of the Babuška-Brezzi theory to each independent equation, lead to the solvability of the continuous and discrete schemes. A reliable and efficient residual-based a posteriori error estimator and its associated adaptive refinement algorithm are introduced and discussed. Five challenging numerical examples are carried out. These give technical details on the implementation of the proposed algorithm and at the same time they show the beauty of the numerical results obtained.
MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35Q79 PDEs in connection with classical thermodynamics and heat transfer
80A17 Thermodynamics of continua
76D05 Navier-Stokes equations for incompressible viscous fluids
76R10 Free convection
Software:
FEniCS; Gmsh
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References:
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