IN THIS PAPER WE PROPOSE AND ANALYZE, UTILIZING MAINLY TOOLS AND ABSTRACT RESULTS FROM BANACH SPACES RATHER THAN FROM HILBERT ONES, A NEW FULLY-MIXED FINITE ELEMENT METHOD FOR THE STATIONARY BOUSSINESQ PROBLEM WITH TEMPERATURE-DEPENDENT VISCOSITY. MORE PRECISELY, FOLLOWING AN IDEA THAT HAS ALREADY BEEN APPLIED TO THE NAVIER-STOKES EQUATIONS AND TO THE FLUID PART ONLY OF OUR MODEL OF INTEREST, WE FIRST INCORPORATE THE VELOCITY GRADIENT AND THE ASSOCIATED BERNOULLI STRESS TENSOR AS AUXILIARY UNKNOWNS. ADDITIONALLY, AND DIFFERENTLY FROM EARLIER WORKS IN WHICH EITHER THE PRIMAL OR THE CLASSICAL DUAL-MIXED METHOD IS EMPLOYED FOR THE HEAT EQUATION, WE CONSIDER HERE AN ANALOGUE OF THE APPROACH FOR THE FLUID, WHICH CONSISTS OF INTRODUCING AS FURTHER VARIABLES THE GRADIENT OF TEMPERATURE AND A VECTOR VERSION OF THE BERNOULLI TENSOR. THE RESULTING MIXED VARIATIONAL FORMULATION, WHICH INVOLVES THE AFOREMENTIONED FOUR UNKNOWNS TOGETHER WITH THE ORIGINAL VARIABLES GIVEN BY THE VELOCITY AND TEMPERATURE OF THE FLUID, IS THEN REFORMULATED AS A FIXED POINT EQUATION. NEXT, WE UTILIZE THE WELL-KNOWN BANACH AND BROUWER THEOREMS, COMBINED WITH THE APPLICATION OF THE BABU$\CHECK{\MATHRM S}$KA-BREZZI THEORY TO EACH INDEPENDENT EQUATION, TO PROVE, UNDER SUITABLE SMALL DATA ASSUMPTIONS, THE EXISTENCE OF A UNIQUE SOLUTION TO THE CONTINUOUS SCHEME, AND THE EXISTENCE OF SOLUTION TO THE ASSOCIATED GALERKIN SYSTEM FOR A FEASIBLE CHOICE OF THE CORRESPONDING FINITE ELEMENT SUBSPACES.

IN THIS PAPER WE INTRODUCE AND ANALYZE A BANACH SPACES-BASED APPROACH YIELDING A FULLY-MIXED FINITE ELEMENT METHOD FOR NUMERICALLY SOLVING THE STATIONARY CHEMOTAXIS-NAVIER-STOKES PROBLEM. THIS IS A NONLINEAR COUPLED MODEL REPRESENTING THE BIOLOGICAL PROCESS GIVEN BY THE CELL MOVEMENT, DRIVEN BY EITHER AN INTERNAL OR AN EXTERNAL CHEMICAL SIGNAL, WITHIN AN INCOMPRESSIBLE FLUID. IN ADDITION TO THE VELOCITY AND PRESSURE OF THE FLUID, THE VELOCITY GRADIENT AND THE BERNOUILLI-TYPE STRESS TENSOR ARE INTRODUCED AS FURTHER UNKNOWNS, WHICH ALLOWS TO ELIMINATE THE PRESSURE FROM THE EQUATIONS AND COMPUTE IT AFTERWARDS VIA A POSTPROCESSING FORMULA. IN TURN, BESIDES THE CELL DENSITY AND THE CHEMICAL SIGNAL CONCENTRATION, THE PSEUDOSTRESS ASSOCIATED WITH THE FORMER AND THE GRADIENT OF THE LATTER ARE INTRODUCED AS AUXILIARY UNKNOWNS AS WELL. THE RESULTING CONTINUOUS FORMULATION, POSED IN SUITABLE BANACH SPACES, CONSISTS OF A COUPLED SYSTEM OF THREE SADDLE POINT-TYPE PROBLEMS, EACH ONE OF THEM PERTURBED WITH TRILINEAR FORMS THAT DEPEND ON DATA AND THE UNKNOWNS OF THE OTHER TWO. THE WELL-POSEDNESS OF IT IS ANALYZED BY MEANS OF A FIXED-POINT STRATEGY, SO THAT THE CLASSICAL BANACH THEOREM, ALONG WITH THE BABU?KA- BREZZI THEORY IN BANACH SPACES, ALLOW TO CONCLUDE, UNDER A SMALLNESS ASSUMPTION ON THE DATA, THE EXISTENCE OF A UNIQUE SOLUTION. ADOPTING AN ANALOGUE APPROACH FOR THE ASSOCIATED GALERKIN SCHEME, AND UNDER SUITABLE HYPOTHESES ON ARBITRARY FINITE ELEMENT SUBSPACES EMPLOYED, WE APPLY THE BROUWER AND BANACH THEOREMS TO SHOW EXISTENCE AND THEN UNIQUENESS OF THE DISCRETE SOLUTION. GENERAL A PRIORI ERROR ESTIMATES, INCLUDING THOSE FOR THE POSTPROCESSED PRESSURE, ARE ALSO DERIVED. NEXT, A SPECIFIC SET OF FINITE ELEMENT SUBSPACES SATISFYING THE REQUIRED STABILITY CONDITIONS, AND YIELDING APPROXIMATE LOCAL CONSERVATION OF MOMENTUM, IS INTRODUCED, WHICH, GIVEN AN INTEGER ? ? 0, IS DEFINED IN TERMS OF RAVIART-THOMAS SPACES OF ORDER ? AND PIECEWISE POLYNOMIALS OF DEGREE ? ? ONLY. THE RE

IN THIS PAPER WE CONSIDER A STRONGLY COUPLED FOW AND NONLINEAR TRANSPORT PROBLEM ARIS ING IN SEDIMENTATION-CONSOLIDATION PROCESSES IN RN , N ? {2, 3} , AND INTRODUCE AND ANA LYZE A BANACH SPACES-BASED VARIATIONAL FORMULATION YIELDING A NEW MIXED-PRIMAL FNITE ELEMENT METHOD FOR ITS NUMERICAL SOLUTION. THE GOVERNING EQUATIONS ARE DETERMINED BY THE COUPLING OF A BRINKMAN FOW WITH A NONLINEAR ADVECTION?DIFUSION EQUATION, IN ADDI TION TO DIRICHLET BOUNDARY CONDITIONS FOR THE FUID VELOCITY AND THE CONCENTRATION. THE APPROACH IS BASED ON THE INTRODUCTION OF THE CAUCHY FUID STRESS AND THE GRADIENT OF ITS VELOCITY AS ADDITIONAL UNKNOWNS, THUS YIELDING A MIXED FORMULATION IN A BANACH SPACES FRAMEWORK FOR THE BRINKMAN EQUATIONS, WHEREAS THE USUAL HILBERTIAN PRIMAL FORMULA TION IS EMPLOYED FOR THE TRANSPORT EQUATION. DIFERENTLY FROM PREVIOUS WORKS ON THIS AND RELATED PROBLEMS, NO AUGMENTED TERMS ARE INCORPORATED, AND HENCE, BESIDES BECOM ING FULLY EQUIVALENT TO THE ORIGINAL PHYSICAL MODEL, THE RESULTING VARIATIONAL FORMULATION IS MUCH SIMPLER, WHICH CONSTITUTES ITS MAIN ADVANTAGE, MAINLY FROM THE COMPUTATIONAL POINT OF VIEW. THE WELL-POSEDNESS OF THE CONTINUOUS FORMULATION IS ANALYZED FRSTLY BY REWRITING IT AS A FXED-POINT OPERATOR EQUATION, AND THEN BY APPLYING THE SCHAUDER AND BANACH THEOREMS, ALONG WITH THE BABU?KA-BREZZI THEORY AND THE LAX-MILGRAM LEMMA. AN ANALOGUE FXED-POINT STRATEGY IS EMPLOYED FOR THE ANALYSIS OF THE ASSOCIATED GALERKIN SCHEME, USING IN THIS CASE THE BROUWER THEOREM INSTEAD OF THE SCHAUDER ONE. THE RESULTING DISCRETE SCHEME BECOMES MOMENTUM CONSERVATIVE FOR THE FUID IN AN APPROXIMATE SENSE. NEXT, A STRANG-TYPE LEMMA AND SUITABLE ALGEBRAIC MANIPULATIONS ARE UTILIZED TO DERIVE THE A PRIORI ERROR ESTIMATES, WHICH, ALONG WITH THE APPROXIMATION PROPERTIES OF THE FNITE ELEMENT SUBSPACES, YIELD THE CORRESPONDING RATES OF CONVERGENCE. THE PAPER IS ENDED WITH SEVERAL NUMERICAL RESULTS ILLUSTRATING THE PERFORMANCE OF THE MIXED-PRIMAL SCHEME AND

IN THIS WORK WE PRESENT AND ANALYZE A FINITE ELEMENT SCHEME YIELDING DISCONTINUOUS GALERKIN APPROXIMATIONS TO THE SOLUTIONS OF THE STATIONARY BOUSSINESQ SYSTEM FOR THE SIMULATION OF NON-ISOTHERMAL FLOW PHENOMENA. THE MODEL CONSISTS OF A NAVIER?STOKES-TYPE SYSTEM, DESCRIBING THE VELOCITY AND THE PRESSURE OF THE FLUID, COUPLED TO AN ADVECTION-DIFFUSION EQUATION FOR THE TEMPERATURE. THE PROPOSED NUMERICAL SCHEME IS BASED ON THE STANDARD INTERIOR PENALTY TECHNIQUE AND AN UPWIND APPROACH FOR THE NONLINEAR CONVECTIVE TERMS AND EMPLOYS THE DIVERGENCE-CONFORMING BREZZI?DOUGLAS?MARINI (BDM) ELEMENTS OF ORDER K FOR THE VELOCITY, DISCONTINUOUS ELEMENTS OF ORDER K-1FOR THE PRESSURE AND DISCONTINUOUS ELEMENTS OF ORDER K FOR THE TEMPERATURE. EXISTENCE AND UNIQUENESS RESULTS ARE SHOWN AND STATED RIGOROUSLY FOR BOTH THE CONTINUOUS PROBLEM AND THE DISCRETE SCHEME, AND OPTIMAL A PRIORI ERROR ESTIMATES ARE ALSO DERIVED. NUMERICAL EXAMPLES BACK UP THE THEORETICAL EXPECTED CONVERGENCE RATES AS WELL AS THE PERFORMANCE OF THE PROPOSED TECHNIQUE.

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.

THIS PAPER PRESENTS OUR CONTRIBUTION TO THE A POSTERIORI ERROR ANALYSIS IN 2D AND 3D OF A SEMI-AUGMENTED MIXED-PRIMAL FINITE ELEMENT METHOD PREVIOUSLY DEVELOPED BY US TO NUMERICALLY SOLVE DOUBLE-DIFFUSIVE NATURAL CONVECTION PROBLEM IN POROUS MEDIA. THE MODEL COMBINES BRINKMAN-NAVIER-STOKES EQUATIONS FOR VELOCITY AND PRESSURE COUPLED TO A VECTOR ADVECTION-DIFFUSION EQUATION, REPRESENTING HEAT AND CONCENTRATION OF A CERTAIN SUBSTANCE IN A VISCOUS FLUID WITHIN A POROUS MEDIUM. STRAIN AND PSEUDO-STRESS TENSORS WERE INTRODUCED TO ESTABLISH SCHEME SOLVABILITY AND PROVIDE A PRIORI ERROR ESTIMATES USING RAVIART-THOMAS ELEMENTS, PIECEWISE POLYNOMIALS AND LAGRANGE FINITE ELEMENTS. IN THIS WORK, WE DERIVE TWO RELIABLE RESIDUAL-BASED A POSTERIORI ERROR ESTIMATORS. THE FIRST ESTIMATOR LEVERAGES ELLIPTICITY PROPERTIES, HELMHOLTZ DECOMPOSITION AS WELL AS CLÉMENT INTERPOLANT AND RAVIART-THOMAS OPERATOR PROPERTIES FOR SHOWING RELIABILITY; EFFICIENCY IS GUARANTEED BY INVERSE INEQUALITIES AND LOCALIZATION STRATEGIES. AN ALTERNATIVE ESTIMATOR IS ALSO DERIVED AND ANALYZED FOR RELIABILITY WITHOUT HELMHOLTZ DECOMPOSITION. NUMERICAL TESTS ARE PRESENTED TO CONFIRM ESTIMATOR PROPERTIES AND DEMONSTRATE ADAPTIVE SCHEME PERFORMANCE.

IN THIS PAPER WE UNDERTAKE AN A POSTERIORI ERROR ANALYSIS ALONG WITH ITS ADAPTIVE COMPUTATION OF A NEW AUGMENTED FULLY-MIXED FINITE ELEMENT METHOD THAT WE HAVE RECENTLY PROPOSED TO NUMERICALLY SIMULATE HEAT DRIVEN FLOWS IN THE BOUSSINESQ APPROXIMATION SETTING. OUR APPROACH INCORPORATES AS ADDITIONAL UNKNOWNS A MODIFIED PSEUDOSTRESS TENSOR FIELD AND AN AUXILIARY VECTOR FIELD IN THE FLUID AND HEAT EQUATIONS, RESPECTIVELY, WHICH POSSIBILITATES THE ELIMINATION OF THE PRESSURE. THIS UNKNOWN, HOWEVER, CAN BE EASILY RECOVERED BY A POSTPROCESSING FORMULA. IN TURN, REDUNDANT GALERKIN TERMS ARE INCLUDED INTO THE WEAK FORMULATION TO ENSURE WELL-POSEDNESS. IN THIS WAY, THE RESULTING VARIATIONAL FORMULATION IS A FOUR-FIELD AUGMENTED SCHEME, WHOSE GALERKIN DISCRETIZATION ALLOWS A RAVIART?THOMAS APPROXIMATION FOR THE AUXILIARY UNKNOWNS AND A LAGRANGE APPROXIMATION FOR THE VELOCITY AND THE TEMPERATURE. IN THE PRESENT WORK, WE PROPOSE A RELIABLE AND EFFICIENT, FULLY-LOCAL AND COMPUTABLE, RESIDUAL-BASED A POSTERIORI ERROR ESTIMATOR IN TWO AND THREE DIMENSIONS FOR THE AFOREMENTIONED METHOD. STANDARD ARGUMENTS BASED ON DUALITY TECHNIQUES, STABLE HELMHOLTZ DECOMPOSITIONS, AND WELL-KNOWN RESULTS FROM PREVIOUS WORKS, ARE THE MAIN UNDERLYING TOOLS USED IN OUR METHODOLOGY. SEVERAL NUMERICAL EXPERIMENTS ILLUSTRATE THE PROPERTIES OF THE ESTIMATOR AND FURTHER VALIDATE THE EXPECTED BEHAVIOR OF THE ASSOCIATED ADAPTIVE ALGORITHM.

IN AN EARLIER WORK OF US, A NEW MIXED FINITE ELEMENT SCHEME WAS DEVELOPED FOR THE BOUSSINESQ MODEL DESCRIBING NATURAL CONVECTION. OUR METHODOLOGY CONSISTED OF A FIXED-POINT STRATEGY FOR THE VARIATIONAL PROBLEM THAT RESULTED AFTER INTRODUCING A MODIFIED PSEUDOSTRESS TENSOR AND THE NORMAL COMPONENT OF THE TEMPERATURE GRADIENT AS AUXILIARY UNKNOWNS IN THE CORRESPONDING NAVIER-STOKES AND ADVECTION-DIFFUSION EQUATIONS DEFINING THE MODEL, RESPECTIVELY, ALONG WITH THE INCORPORATION OF PARAMETERIZED REDUNDANT GALERKIN TERMS. THE WELL-POSEDNESS OF BOTH THE CONTINUOUS AND DISCRETE SETTINGS, THE CONVERGENCE OF THE ASSOCIATED GALERKIN SCHEME, AS WELL AS A PRIORI ERROR ESTIMATES OF OPTIMAL ORDER WERE STATED THERE. IN THIS WORK WE COMPLEMENT THE NUMERICAL ANALYSIS OF OUR AFOREMENTIONED AUGMENTED MIXED-PRIMAL METHOD BY CARRYING OUT A CORRESPONDING A POSTERIORI ERROR ESTIMATION IN TWO AND THREE DIMENSIONS. STANDARD ARGUMENTS RELYING ON DUALITY TECHNIQUES, AND SUITABLE HELMHOLTZ DECOMPOSITIONS ARE USED TO DERIVE A GLOBAL ERROR INDICATOR AND TO SHOW ITS RELIABILITY. A GLOBALLY EFFICIENCY PROPERTY WITH RESPECT TO THE NATURAL NORM IS FURTHER PROVED VIA USUAL LOCALIZATION TECHNIQUES OF BUBBLE FUNCTIONS. FINALLY, AN ADAPTIVE ALGORITHM BASED ON A RELIABLE, FULLY LOCAL AND COMPUTABLE A POSTERIORI ERROR ESTIMATOR INDUCED BY THE AFOREMENTIONED ONE IS PROPOSED, AND ITS PERFORMANCE AND EFFECTIVENESS ARE ILLUSTRATED THROUGH A FEW NUMERICAL EXAMPLES IN TWO DIMENSIONS.

IN THIS PAPER WE PROPOSE AND ANALYZE A NEW FULLY-MIXED FINITE ELEMENT METHOD FOR THE STATIONARY BOUSSINESQ PROBLEM. MORE PRECISELY, WE REFORMULATE A PREVIOUS PRIMAL-MIXED SCHEME FOR THE RESPECTIVE MODEL BY HOLDING THE SAME MODIFIED PSEUDOSTRESS TENSOR DEPENDING ON THE PRESSURE, AND THE DIFFUSIVE AND CONVECTIVE TERMS OF THE NAVIER?STOKES EQUATIONS FOR THE FLUID; AND IN CONTRAST, WE NOW INTRODUCE A NEW AUXILIARY VECTOR UNKNOWN INVOLVING THE TEMPERATURE, ITS GRADIENT AND THE VELOCITY FOR THE HEAT EQUATION. AS A CONSEQUENCE, A MIXED APPROACH IS CARRIED OUT IN HEAT AS WELL AS FLUID EQUATION, AND DIFFERENTLY FROM THE PREVIOUS SCHEME, NO BOUNDARY UNKNOWNS ARE NEEDED, WHICH LEADS TO AN IMPROVEMENT OF THE METHOD FROM BOTH THE THEORETICAL AND COMPUTATIONAL POINT OF VIEW. IN FACT, THE PRESSURE IS ELIMINATED AND AS A RESULT THE UNKNOWNS ARE GIVEN BY THE AFOREMENTIONED AUXILIARY VARIABLES, THE VELOCITY AND THE TEMPERATURE OF THE FLUID. IN ADDITION, FOR REASONS OF SUITABLE REGULARITY CONDITIONS, THE SCHEME IS AUGMENTED BY USING THE CONSTITUTIVE AND EQUILIBRIUM EQUATIONS, AND THE DIRICHLET BOUNDARY CONDITIONS. THEN, THE RESULTING FORMULATION IS REWRITTEN AS A FIXED POINT PROBLEM AND ITS WELL-POSEDNESS IS GUARANTEED BY THE CLASSICAL BANACH THEOREM COMBINED WITH THE LAX?MILGRAM THEOREM. AS FOR THE ASSOCIATED GALERKIN SCHEME, THE BROUWER AND THE BANACH FIXED POINT THEOREMS ARE UTILIZED TO ESTABLISH EXISTENCE AND UNIQUENESS OF DISCRETE SOLUTION, RESPECTIVELY. IN PARTICULAR, RAVIART?THOMAS SPACES OF ORDER K FOR THE AUXILIARY UNKNOWNS AND CONTINUOUS PIECEWISE POLYNOMIALS OF DEGREE ?K+1 FOR THE VELOCITY AND THE TEMPERATURE BECOME FEASIBLE CHOICES. FINALLY, WE DERIVE OPTIMAL A PRIORI ERROR ESTIMATES AND PROVIDE SEVERAL NUMERICAL RESULTS ILLUSTRATING THE GOOD PERFORMANCE OF THE SCHEME AND CONFIRMING THE THEORETICAL RATES OF CONVERGENCE.

IN THIS PAPER WE STUDY A STATIONARY DOUBLE-DIFFUSIVE NATURAL CONVECTION PROBLEM IN POROUS MEDIA GIVEN BY A NAVIER-STOKES/BRINKMAN TYPE SYSTEM, FOR DESCRIBING THE VELOCITY AND THE PRESSURE, COUPLED TO A VECTOR ADVECTION-DIFFUSION EQUATION RELATE TO THE HEAT AND SUBSTANCE CONCENTRATION, OF A VISCOUS FLUID IN A POROUS MEDIA WITH PHYSICAL BOUNDARY CONDITIONS. THE MODEL PROBLEM IS REWRITTEN IN TERMS OF A FIRST-ORDER SYSTEM, WITHOUT THE PRESSURE, BASED ON THE INTRODUCTION OF THE STRAIN TENSOR AND A NONLINEAR PSEUDO-STRESS TENSOR IN THE FLUID EQUATIONS. AFTER A VARIATIONAL APPROACH, THE RESULTING WEAK MODEL IS THEN AUGMENTED USING APPROPRIATE REDUNDANT PENALIZATION TERMS FOR THE FLUID EQUATIONS ALONG WITH A STANDARD PRIMAL FORMULATION FOR THE HEAT AND SUBSTANCE CONCENTRATION. THEN, IT IS REWRITTEN AS AN EQUIVALENT FIXED-POINT PROBLEM. WELL-POSEDNESS RESULTS FOR BOTH THE CONTINUOUS AND THE DISCRETE SCHEMES ARE STATED, AS WELL AS THE RESPECTIVE CONVERGENCE RESULT UNDER CERTAIN REGULARITY ASSUMPTIONS COMBINED WITH THE LAX-MILGRAM THEOREM, AND THE BANACH AND BROUWER FIXED-POINT THEOREMS. IN PARTICULAR, RAVIART-THOMAS ELEMENTS OF ORDER K ARE USED FOR APPROXIMATING THE PSEUDO-STRESS TENSOR, PIECEWISE POLYNOMIALS OF DEGREE ?K AND ARE UTILIZED FOR APPROXIMATING THE STRAIN TENSOR AND THE VELOCITY, RESPECTIVELY, AND THE HEAT AND SUBSTANCE CONCENTRATION ARE APPROXIMATED BY MEANS OF LAGRANGE FINITE ELEMENTS OF ORDER . OPTIMAL A PRIORI ERROR ESTIMATES ARE DERIVED AND CONFIRMED THROUGH SOME NUMERICAL EXAMPLES THAT ILLUSTRATE THE PERFORMANCE OF THE PROPOSED SEMI-AUGMENTED MIXED-PRIMAL SCHEME.

IN THIS PAPER WE STUDY A STATIONARY GENERALIZED BIOCONVECTION PROBLEM GIVEN BY A NAVIER?STOKES TYPE SYSTEM COUPLED TO A CELL CONSERVATION EQUATION FOR DESCRIBING THE HYDRODYNAMIC AND MICRO-ORGANISMS CONCENTRATION, RESPECTIVELY, OF A CULTURE FLUID, ASSUMED TO BE VISCOUS AND INCOMPRESSIBLE, AND IN WHICH THE VISCOSITY DEPENDS ON THE CONCENTRATION. THE MODEL IS REWRITTEN IN TERMS OF A FIRST-ORDER SYSTEM BASED ON THE INTRODUCTION OF THE SHEAR-STRESS, THE VORTICITY, AND THE PSEUDO-STRESS TENSORS IN THE FLUID EQUATIONS ALONG WITH AN AUXILIARY VECTOR IN THE CONCENTRATION EQUATION. AFTER A VARIATIONAL APPROACH, THE RESULTING WEAK MODEL IS THEN AUGMENTED USING APPROPRIATE REDUNDANT PARAMETERIZED TERMS AND REWRITTEN AS FIXED-POINT PROBLEM. EXISTENCE AND UNIQUENESS RESULTS FOR BOTH THE CONTINUOUS AND THE DISCRETE SCHEME AS WELL AS THE RESPECTIVE CONVERGENCE RESULT ARE OBTAINED UNDER CERTAIN REGULARITY ASSUMPTIONS COMBINED WITH THE LAX?MILGRAM THEOREM, AND THE BANACH AND BROUWER FIXED-POINT THEOREMS. OPTIMAL A PRIORI ERROR ESTIMATES ARE DERIVED AND CONFIRMED THROUGH SOME NUMERICAL EXAMPLES THAT ILLUSTRATE THE PERFORMANCE OF THE PROPOSED TECHNIQUE.

IN THIS ARTICLE, WE PROPOSE AND ANALYZE A NEW MIXED VARIATIONAL FORMULATION FOR THE STATIONARY BOUSSINESQ PROBLEM. OUR METHOD, WHICH USES A TECHNIQUE PREVIOUSLY APPLIED TO THE NAVIER?STOKES EQUATIONS, IS BASED FIRST ON THE INTRODUCTION OF A MODIFIED PSEUDOSTRESS TENSOR DEPENDING NONLINEARLY ON THE VELOCITY THROUGH THE RESPECTIVE CONVECTIVE TERM. NEXT, THE PRESSURE IS ELIMINATED, AND AN AUGMENTED APPROACH FOR THE FLUID FLOW, WHICH INCORPORATES GALERKIN-TYPE TERMS ARISING FROM THE CONSTITUTIVE AND EQUILIBRIUM EQUATIONS, AND FROM THE DIRICHLET BOUNDARY CONDITION, IS COUPLED WITH A PRIMAL-MIXED SCHEME FOR THE MAIN EQUATION MODELING THE TEMPERATURE. IN THIS WAY, THE ONLY UNKNOWNS OF THE RESULTING FORMULATION ARE GIVEN BY THE AFOREMENTIONED NONLINEAR PSEUDOSTRESS, THE VELOCITY, THE TEMPERATURE, AND THE NORMAL DERIVATIVE OF THE LATTER ON THE BOUNDARY. AN EQUIVALENT FIXED-POINT SETTING IS THEN INTRODUCED AND THE CORRESPONDING CLASSICAL BANACH THEOREM, COMBINED WITH THE LAX?MILGRAM THEOREM AND THE BABU?KA?BREZZI THEORY, ARE APPLIED TO PROVE THE UNIQUE SOLVABILITY OF THE CONTINUOUS PROBLEM. IN TURN, THE BROUWER AND THE BANACH FIXED-POINT THEOREMS ARE USED TO ESTABLISH EXISTENCE AND UNIQUENESS OF SOLUTION, RESPECTIVELY, OF THE ASSOCIATED GALERKIN SCHEME. IN PARTICULAR, RAVIART?THOMAS SPACES OF ORDER K FOR THE PSEUDOSTRESS, CONTINUOUS PIECEWISE POLYNOMIALS OF DEGREE ? K+1 FOR THE VELOCITY AND THE TEMPERATURE, AND PIECEWISE POLYNOMIALS OF DEGREE ? K FOR THE BOUNDARY UNKNOWN BECOME FEASIBLE CHOICES. FINALLY, WE DERIVE OPTIMAL A PRIORI ERROR ESTIMATES, AND PROVIDE SEVERAL NUMERICAL RESULTS ILLUSTRATING THE GOOD PERFORMANCE OF THE AUGMENTED MIXED-PRIMAL FINITE ELEMENT METHOD AND CONFIRMING THE THEORETICAL RATES OF CONVERGENCE.