Publicación: A BANACH SPACES-BASED MIXED-PRIMAL FINITE ELEMENT METHOD FOR THE COUPLING OF BRINKMAN FLOW AND NONLINEAR TRANSPORT
Fecha
2022
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CALCOLO
Resumen
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
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NONLIMEAR TRANSPORT PROBLEM, FIXED POINT THEORY, FINITY ELEMENT METHODS, BRINKMAN EQUATIONS, A PRIORI ERROR ANALYSIS
