Publicación: A FULLY-MIXED FINITE ELEMENT METHOD FOR THE STEADY STATE OBERBECK-BOUSSINESQ SYSTEM
Fecha
2020
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SMAI JOURNAL OF COMPUTATIONAL MATHEMATICS
Resumen
WE PROPOSE A NEW FULLY-MIXED FORMULATION FOR THE STATIONARY OBERBECK-BOUSSINESQ PROBLEM WHEN
VISCOSITY DEPENDS ON BOTH TEMPERATURE AND CONCENTRATION. FOLLOWING SIMILAR IDEAS APPLIED PRE-
VIOUSLY TO THE BOUSSINESQ AND NAVIER-STOKES EQUATIONS, WE INCORPORATE THE VELOCITY GRADIENT AND
THE BERNOULLI STRESS TENSOR AS AUXILIARY UNKNOWNS OF THE FLUID EQUATIONS. IN TURN, THE GRADIENTS
OF TEMPERATURE AND OF CONCENTRATION, IN ADDITION TO A BERNOULLI VECTOR, ARE INTRODUCED AS FURTHER
VARIABLES OF THE HEAT AND MASS TRANSFER EQUATIONS. CONSEQUENTLY, A DUAL-MIXED APPROACH WITH
DIRICHLET DATA IS DEFINED IN EACH SUB-SYSTEM, AND THE WELL-KNOWN BANACH AND BROUWER THEOREMS
ARE COMBINED WITH BABU?SKA-BREZZI?S THEORY IN EACH INDEPENDENT SET OF EQUATIONS, YIELDING THE SOLV-
ABILITY OF THE CONTINUOUS AND DISCRETE SCHEMES. NEXT, WE DESCRIBE SPECIFIC FINITE ELEMENT SUBSPACES
SATISFYING APPROPRIATE STABILITY REQUIREMENTS, AND DERIVE OPTIMAL A PRIORI ERROR ESTIMATES. FINALLY,
SEVERAL NUMERICAL EXAMPLES ILLUSTRATING THE PERFORMANCE OF THE FULLY-MIXED SCHEME AND CONFIRMING
THE THEORETICAL RATES OF CONVERGENCE ARE PRESENTED.
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Palabras clave
Oberbeck?Boussinesq equations, fully?mixed formulation, fixed-point theory, finite element methods, a priori error analysis
