IN THIS PAPER WE DEVELOP THE A PRIORI ANALYSIS OF A MIXED FINITE ELEMENT METHOD FOR THE COUPLING OF FLUID FLOW WITH POROUS MEDIA FLOW. FLOWS ARE GOVERNED BY THE NAVIER?STOKES AND DARCY EQUATIONS, RESPECTIVELY, AND THE CORRESPONDING TRANSMISSION CONDITIONS ARE GIVEN BY MASS CONSERVATION, BALANCE OF NORMAL FORCES, AND THE BEAVERS-JOSEPH-SAFFMAN LAW. WE CONSIDER THE STANDARD MIXED FORMULATION IN THE NAVIER?STOKES DOMAIN AND THE DUAL-MIXED ONE IN THE DARCY REGION, WHICH YIELDS THE INTRODUCTION OF THE TRACE OF THE POROUS MEDIUM PRESSURE AS A SUITABLE LAGRANGE MULTIPLIER. THE FINITE ELEMENT SUBSPACES DEFINING THE DISCRETE FORMULATION EMPLOY BERNARDI-RAUGEL AND RAVIART-THOMAS ELEMENTS FOR THE VELOCITIES, PIECEWISE CONSTANTS FOR THE PRESSURES, AND CONTINUOUS PIECEWISE LINEAR ELEMENTS FOR THE LAGRANGE MULTIPLIER. WE SHOW STABILITY, CONVERGENCE, AND A PRIORI ERROR ESTIMATES FOR THE ASSOCIATED GALERKIN SCHEME. FINALLY, SEVERAL NUMERICAL RESULTS ILLUSTRATING THE GOOD PERFORMANCE OF THE METHOD AND CONFIRMING THE THEORETICAL RATES OF CONVERGENCE ARE REPORTED.

IN THIS PAPER WE DEVELOP THE A PRIORI ANALYSIS OF A MIXED FINITE ELEMENT METHOD FOR THE COUPLING OF FLUID FLOW WITH POROUS MEDIA FLOW. FLOWS ARE GOVERNED BY THE NAVIER-STOKES AND DARCY EQUATIONS, RESPECTIVELY, AND THE CORRESPONDING TRANSMISSION CONDITIONS ARE GIVEN BY MASS CONSERVATION, BALANCE OF NORMAL FORCES, AND THE BEAVERS-JOSEPH-SAFFMAN LAW. WE CONSIDER THE STANDARD MIXED FORMULATION IN THE NAVIER?STOKES DOMAIN AND THE DUAL-MIXED ONE IN THE DARCY REGION, WHICH YIELDS THE INTRODUCTION OF THE TRACE OF THE POROUS MEDIUM PRESSURE AS A SUITABLE LAGRANGE MULTIPLIER. THE FINITE ELEMENT SUBSPACES DEFINING THE DISCRETE FORMULATION EMPLOY BERNARDI-RAUGEL AND RAVIART-THOMAS ELEMENTS FOR THE VELOCITIES, PIECEWISE CONSTANTS FOR THE PRESSURES, AND CONTINUOUS PIECEWISE LINEAR ELEMENTS FOR THE LAGRANGE MULTIPLIER. WE SHOW STABILITY, CONVERGENCE, AND A PRIORI ERROR ESTIMATES FOR THE ASSOCIATED GALERKIN SCHEME. FINALLY, SEVERAL NUMERICAL RESULTS ILLUSTRATING THE GOOD PERFORMANCE OF THE METHOD AND CONFIRMING THE THEORETICAL RATES OF CONVERGENCE ARE REPORTED.

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.

IN THIS WORK WE PROPOSE AND ANALYZE A NEW FULLY DIVERGENCE-CONFORMING FINITE ELEMENT METHOD FOR THE NUMERICAL SIMULATION OF THE BOUSSINESQ PROBLEM, DESCRIBING THE MOTION OF A NON-ISOTHERMAL INCOMPRESSIBLE FLUID SUBJECT TO A HEAT SOURCE. WE CONSIDER THE STANDARD VELOCITY-PRESSURE FORMULATION FOR THE FLUID FLOW EQUATION AND THE DUAL-MIXED ONE FOR THE HEAT EQUATION. IN THIS WAY, THE UNKNOWNS OF THE RESULTING FORMULATION ARE GIVEN BY THE VELOCITY, THE PRESSURE, THE TEMPERATURE AND THE GRADIENT OF THE LATTER. THE CORRESPONDING GALERKIN SCHEME MAKES USE OF A NONCONFORMING EXACTLY DIVERGENCE-FREE APPROACH TO APPROXIMATE THE VELOCITY AND PRESSURE, AND EMPLOYS STANDARD H(DIV)-CONFORMING ELEMENTS FOR THE GRADIENT OF THE TEMPERATURE AND DISCONTINUOUS ELEMENTS FOR THE TEMPERATURE. SINCE HERE WE UTILIZE A DUAL-MIXED FORMULATION FOR THE HEAT EQUATION, THE TEMPERATURE DIRICHLET BOUNDARY CONDITION BECOMES NATURAL, THUS THERE IS NO NEED OF INTRODUCING A SUFFICIENTLY SMALL DISCRETE LIFTING TO PROVE WELL-POSEDNESS OF THE DISCRETE PROBLEM. MOREOVER, THE RESULTING NUMERICAL SCHEME YIELDS EXACTLY DIVERGENCE-FREE VELOCITY APPROXIMATIONS; THUS, IT IS PROBABLY ENERGY-STABLE WITHOUT THE NEED OF MODIFYING THE UNDERLYING DIFFERENTIAL EQUATIONS, AND PROVIDE AN OPTIMAL CONVERGENT APPROXIMATION OF THE TEMPERATURE GRADIENT. THE ANALYSIS OF THE CONTINUOUS AND DISCRETE PROBLEMS IS CARRIED OUT BY MEANS OF A FIXED-POINT STRATEGY UNDER A SUFFICIENTLY SMALL DATA ASSUMPTION. WE DERIVE OPTIMAL ERROR ESTIMATES IN THE MESH SIZE FOR SMOOTH SOLUTIONS AND PROVIDE SEVERAL NUMERICAL RESULTS ILLUSTRATING THE PERFORMANCE OF THE METHOD AND CONFIRMING THE THEORETICAL RATES OF CONVERGENCE.

IN THIS WORK WE INTRODUCE AND ANALYZE A NEW AUGMENTED FULLY-MIXED FORMULATION FOR THE STATIONARY NAVIER?STOKES/DARCY COUPLED PROBLEM. OUR APPROACH EMPLOYS, ON THE FREE-FLUID REGION, A TECHNIQUE PREVIOUSLY APPLIED TO THE STATIONARY NAVIER?STOKES EQUATIONS, WHICH CONSISTS OF THE INTRODUCTION OF A MODIFIED PSEUDOSTRESS TENSOR INVOLVING THE DIFFUSIVE AND CONVECTIVE TERMS, TOGETHER WITH THE PRESSURE. IN ADDITION, BY USING THE INCOMPRESSIBILITY CONDITION, THE PRESSURE IS ELIMINATED, AND SINCE THE CONVECTIVE TERM FORCES THE FREE-FLUID VELOCITY TO LIVE IN A SMALLER SPACE THAN USUAL, WE AUGMENT THE RESULTING FORMULATION WITH SUITABLE GALERKIN TYPE TERMS ARISING FROM THE CONSTITUTIVE AND EQUILIBRIUM EQUATIONS. ON THE OTHER HAND, IN THE DARCY REGION WE APPLY THE USUAL DUAL-MIXED FORMULATION, WHICH YIELDS THE INTRODUCTION OF THE TRACE OF THE POROUS MEDIA PRESSURE AS AN ASSOCIATED LAGRANGE MULTIPLIER. THE LATTER IS CONNECTED WITH THE FACT THAT ONE OF THE TRANSMISSION CONDITIONS INVOLVING MASS CONSERVATION BECOMES ESSENTIAL AND MUST BE IMPOSED WEAKLY. IN THIS WAY, WE OBTAIN A FIVE-FIELD FORMULATION WHERE THE PSEUDOSTRESS AND THE VELOCITY IN THE FLUID, TOGETHER WITH THE VELOCITY AND THE PRESSURE IN THE POROUS MEDIUM, AND THE AFOREMENTIONED LAGRANGE MULTIPLIER, ARE THE CORRESPONDING UNKNOWNS. THE WELL-POSEDNESS ANALYSIS IS CARRIED OUT BY COMBINING THE CLASSICAL BABU?KA?BREZZI THEORY AND THE BANACH FIXED-POINT THEOREM. A PROPER ADAPTATION OF THE ARGUMENTS EXPLOITED IN THE CONTINUOUS ANALYSIS ALLOWS US TO STATE SUITABLE HYPOTHESES ON THE FINITE ELEMENT SUBSPACES ENSURING THAT THE ASSOCIATED GALERKIN SCHEME IS WELL-POSED AND CONVERGENT. IN PARTICULAR, RAVIART?THOMAS ELEMENTS OF LOWEST ORDER FOR THE PSEUDOSTRESS AND THE DARCY VELOCITY, CONTINUOUS PIECEWISE LINEAR POLYNOMIALS FOR THE FREE-FLUID VELOCITY, PIECEWISE CONSTANTS FOR THE DARCY PRESSURE, TOGETHER WITH CONTINUOUS PIECEWISE LINEAR ELEMENTS FOR THE LAGRANGE MULTIPLIER, CONSTITUTE FEASIBLE CHOICES. FINALLY, WE PROVIDE SEVERAL NUME

WE PROPOSE AND ANALYSE AN AUGMENTED MIXED FINITE ELEMENT METHOD FOR THE COUPLING OF FLUID FLOW WITH POROUS MEDIA FLOW. THE FLOWS ARE GOVERNED BY A CLASS OF NONLINEAR NAVIER?STOKES AND LINEAR DARCY EQUATIONS, RESPECTIVELY, AND THE TRANSMISSION CONDITIONS ARE GIVEN BY MASS CONSERVATION, BALANCE OF NORMAL FORCES, AND THE BEAVERS?JOSEPH?SAFFMAN LAW. WE APPLY DUAL-MIXED FORMULATIONS IN BOTH DOMAINS, AND THE NONLINEARITY INVOLVED IN THE NAVIER?STOKES REGION IS HANDLED BY SETTING THE STRAIN AND VORTICITY TENSORS AS AUXILIARY UNKNOWNS. IN TURN, SINCE THE TRANSMISSION CONDITIONS BECOME ESSENTIAL, THEY ARE IMPOSED WEAKLY, WHICH YIELDS THE INTRODUCTION OF THE TRACES OF THE POROUS MEDIA PRESSURE AND THE FLUID VELOCITY AS THE ASSOCIATED LAGRANGE MULTIPLIERS. FURTHERMORE, SINCE THE CONVECTIVE TERM IN THE FLUID FORCES THE VELOCITY TO LIVE IN A SMALLER SPACE THAN USUAL, WE AUGMENT THE VARIATIONAL FORMULATION WITH SUITABLE GALERKIN TYPE TERMS ARISING FROM THE CONSTITUTIVE AND EQUILIBRIUM EQUATIONS OF THE NAVIER?STOKES EQUATIONS, AND THE RELATION DEFINING THE STRAIN AND VORTICITY TENSORS. THE RESULTING AUGMENTED SCHEME IS THEN WRITTEN EQUIVALENTLY AS A FIXED POINT EQUATION, SO THAT THE WELL-KNOWN SCHAUDER AND BANACH THEOREMS, COMBINED WITH CLASSICAL RESULTS ON BIJECTIVE MONOTONE OPERATORS, ARE APPLIED TO PROVE THE UNIQUE SOLVABILITY OF THE CONTINUOUS AND DISCRETE SYSTEMS. IN PARTICULAR, GIVEN AN INTEGER K ? 0, PIECEWISE POLYNOMIALS OF DEGREE

WE PROPOSE AND ANALYZE A NEW MIXED FINITE ELEMENT METHOD FOR THE PROBLEM OF STEADY DOUBLE-DIFFUSIVE CONVECTION IN A FLUID-SATURATED POROUS MEDIUM. MORE PRECISELY, THE MODEL IS DESCRIBED BY THE COUPLING OF THE BRINKMAN?FORCHHEIMER AND DOUBLE-DIFFUSION EQUATIONS, IN WHICH THE ORIGINALLY SOUGHT VARIABLES ARE THE VELOCITY AND PRESSURE OF THE FLUID, AND THE TEMPERATURE AND CONCENTRATION OF A SOLUTE. OUR APPROACH IS BASED ON THE INTRODUCTION OF THE FURTHER UNKNOWNS GIVEN BY THE FLUID PSEUDOSTRESS TENSOR, AND THE PSEUDOHEAT AND PSEUDODIFFUSIVE VECTORS, THUS YIELDING A FULLY-MIXED FORMULATION. FURTHERMORE, SINCE THE NONLINEAR TERM IN THE BRINKMAN?FORCHHEIMER EQUATION REQUIRES THE VELOCITY TO LIVE IN A SMALLER SPACE THAN USUAL, WE PARTIALLY AUGMENT THE VARIATIONAL FORMULATION WITH SUITABLE GALERKIN TYPE TERMS, WHICH FORCES BOTH THE TEMPERATURE AND CONCENTRATION SCALAR FIELDS TO LIVE IN L4. AS A CONSEQUENCE, THE AFOREMENTIONED PSEUDOHEAT AND PSEUDODIFFUSIVE VECTORS LIVE IN A SUITABLE H(DIV)-TYPE BANACH SPACE. THE RESULTING AUGMENTED SCHEME IS WRITTEN EQUIVALENTLY AS A FIXED POINT EQUATION, SO THAT THE WELL-KNOWN SCHAUDER AND BANACH THEOREMS, COMBINED WITH THE LAX?MILGRAM AND BANACH?NE?AS?BABU?KA THEOREMS, ALLOW TO PROVE THE UNIQUE SOLVABILITY OF THE CONTINUOUS PROBLEM. AS FOR THE ASSOCIATED GALERKIN SCHEME WE UTILIZE RAVIART?THOMAS SPACES OF ORDER K?0 FOR APPROXIMATING THE PSEUDOSTRESS TENSOR, AS WELL AS THE PSEUDOHEAT AND PSEUDODIFFUSIVE VECTORS, WHEREAS CONTINUOUS PIECEWISE POLYNOMIALS OF DEGREE ?K+1 ARE EMPLOYED FOR THE VELOCITY, AND PIECEWISE POLYNOMIALS OF DEGREE ?K FOR THE TEMPERATURE AND CONCENTRATION FIELDS. IN TURN, THE EXISTENCE AND UNIQUENESS OF THE DISCRETE SOLUTION IS ESTABLISHED SIMILARLY TO ITS CONTINUOUS COUNTERPART, APPLYING IN THIS CASE THE BROUWER AND BANACH FIXED-POINT THEOREMS, RESPECTIVELY. FINALLY, WE DERIVE OPTIMAL A PRIORI ERROR ESTIMATES AND PROVIDE SEVERAL NUMERICAL RESULTS CONFIRMING THE THEORETICAL RATES OF CONVERGENCE AND ILLUSTRATING THE PERFORM

WE PROPOSE AND ANALYZE A HIGH ORDER MIXED FINITE ELEMENT METHOD FOR DIFFUSION PROBLEMS WITH DIRICHLET BOUNDARY CONDITION ON A DOMAIN OMEGA WITH CURVED BOUNDARY GAMMA. THE METHOD IS BASED ON APPROXIMATING OMEGA BY A POLYGONAL SUBDOMAIN D-H, WITH BOUNDARY GAMMA(H), WHERE A HIGH ORDER CONFORMING GALERKIN METHOD IS CONSIDERED TO COMPUTE THE SOLUTION. TO APPROXIMATE THE DIRICHLET DATA ON THE COMPUTATIONAL BOUNDARY GAMMA(H), WE EMPLOY A TRANSFERRING TECHNIQUE BASED ON INTEGRATING THE EXTRAPOLATED DISCRETE GRADIENT ALONG SEGMENTS JOINING GAMMA(H) AND GAMMA. CONSIDERING GENERAL FINITE DIMENSIONAL SUBSPACES WE PROVE THAT THE RESULTING GALERKIN SCHEME, WHICH IS H(DIV; D-H)-CONFORMING, IS WELL-POSED PROVIDED SUITABLE HYPOTHESES ON THE AFOREMENTIONED SUBSPACES AND INTEGRATION SEGMENTS. A FEASIBLE CHOICE OF DISCRETE SPACES IS GIVEN BY RAVIART-THOMAS ELEMENTS OF ORDER K >= 0 FOR THE VECTORIAL VARIABLE AND DISCONTINUOUS POLYNOMIALS OF DEGREE K FOR THE SCALAR VARIABLE, YIELDING OPTIMAL CONVERGENCE IF THE DISTANCE BETWEEN GAMMA(H) AND GAMMA IS AT MOST OF THE ORDER OF THE MESHSIZE H. WE ALSO APPROXIMATE THE SOLUTION IN D-H(C) := OMEGA\(D-H) OVER BAR AND DERIVE THE CORRESPONDING ERROR ESTIMATES. NUMERICAL EXPERIMENTS ILLUSTRATE THE PERFORMANCE OF THE SCHEME AND VALIDATE THE THEORY.

IN THIS PAPER WE FOCUS ON THE ANALYSIS OF A MIXED FINITE ELEMENT METHOD FOR A CLASS OF NATURAL CONVECTION PROBLEMS IN TWO DIMENSIONS. MORE PRECISELY, WE CONSIDER A SYSTEM BASED ON THE COUPLING OF THE STEADY-STATE EQUATIONS OF MOMENTUM (NAVIER?STOKES) AND THERMAL ENERGY BY MEANS OF THE BOUSSINESQ APPROXIMATION (COINED THE BOUSSINESQ PROBLEM), WHERE WE ALSO TAKE INTO ACCOUNT A TEMPERATURE DEPENDENCE OF THE VISCOSITY OF THE FLUID. THE CONSTRUCTION OF THIS FINITE ELEMENT METHOD BEGINS WITH THE INTRODUCTION OF THE PSEUDOSTRESS AND VORTICITY TENSORS, AND A MIXED FORMULATION FOR THE MOMENTUM EQUATIONS, WHICH IS AUGMENTED WITH GALERKIN-TYPE TERMS, IN ORDER TO DEAL WITH THE NON-LINEARITY OF THESE EQUATIONS AND THE CONVECTIVE TERM IN THE ENERGY EQUATION, WHERE A PRIMAL FORMULATION IS CONSIDERED. THE PRESCRIBED TEMPERATURE ON THE BOUNDARY BECOMES AN ESSENTIAL CONDITION, WHICH IS WEAKLY IMPOSED, LEADING US TO THE DEFINITION OF THE NORMAL HEAT FLUX THROUGH THE BOUNDARY AS A LAGRANGE MULTIPLIER. WE SHOW THAT THIS HIGHLY COUPLED PROBLEM CAN BE UNCOUPLED AND ANALYSED AS A FIXED-POINT PROBLEM, WHERE BANACH AND BROUWER THEOREMS WILL HELP US TO PROVIDE SUFFICIENT CONDITIONS TO ENSURE WELL-POSEDNESS OF THE PROBLEMS ARISING FROM THE CONTINUOUS AND DISCRETE FORMULATIONS, ALONG WITH SEVERAL APPLICATIONS OF CONTINUOUS INJECTIONS GUARANTEED BY THE RELLICH?KONDRACHOV THEOREM. FINALLY, WE SHOW SOME NUMERICAL RESULTS TO ILLUSTRATE THE PERFORMANCE OF THIS FINITE ELEMENT METHOD, AS WELL AS TO PROVE THE ASSOCIATED RATES OF CONVERGENCE.

IN THIS PAPER WE PROPOSE A NEW MIXED-PRIMAL FORMULATION FOR HEAT-DRIVEN FLOWS WITH TEMPERATURE-DEPENDENT VISCOSITY MODELED BY THE STATIONARY BOUSSINESQ EQUATIONS. WE ANALYZE THE WELL-POSEDNESS OF THE GOVERNING EQUATIONS IN THIS MATHEMATICAL STRUCTURE, FOR WHICH WE EMPLOY THE BANACH FIXED-POINT THEOREM AND THE GENERALIZED THEORY OF SADDLE-POINT PROBLEMS. THE MOTIVATION IS TO OVERCOME A DRAWBACK IN A RECENT WORK BY THE AUTHORS WHERE, IN THE MIXED FORMULATION FOR THE MOMENTUM EQUATION, THE RECIPROCAL OF THE VISCOSITY IS A PRE-FACTOR TO A TENSOR PRODUCT OF VELOCITIES; MAKING THE ANALYSIS QUITE RESTRICTIVE, AS ONE NEEDS A GIVEN CONTINUOUS INJECTION THAT HOLDS ONLY IN 2D. WE SHOW IN THIS WORK THAT BY ADDING BOTH THE PSEUDO-STRESS AND THE STRAIN RATE TENSORS AS NEW UNKNOWNS IN THE PROBLEM, WE GET MORE FLEXIBILITY IN THE ANALYSIS, COVERING ALSO THE 3D CASE. THE REST OF THE FORMULATION IS BASED ON ELIMINATING THE PRESSURE, INCORPORATING AUGMENTED GALERKIN-TYPE TERMS IN THE MIXED FORM OF THE MOMENTUM EQUATION, AND DEFINING THE NORMAL HEAT FLUX AS A SUITABLE LAGRANGE MULTIPLIER IN A PRIMAL FORMULATION FOR THE ENERGY EQUATION. ADDITIONALLY, THE SYMMETRY OF THE STRESS IS IMPOSED IN AN ULTRA-WEAK SENSE, AND CONSEQUENTLY THE VORTICITY TENSOR IS NO LONGER REQUIRED AS PART OF THE UNKNOWNS. A FINITE ELEMENT METHOD THAT FOLLOWS THE SAME SETTING IS THEN PROPOSED, WHERE WE REMARK THAT BOTH PRESSURE AND VORTICITY CAN BE RECOVERED FROM THE PRINCIPAL UNKNOWNS VIA POSTPROCESSING FORMULAE. THE SOLVABILITY OF THE DISCRETE PROBLEM IS ANALYZED BY MEANS OF THE BROUWER FIXED-POINT THEOREM, AND WE DERIVE ERROR ESTIMATES IN SUITABLE NORMS. NUMERICAL EXAMPLES ILLUSTRATE THE PERFORMANCE OF THE NEW SCHEM AND ITS USE IN THE SIMULATION OF MANTLE CONVECTION, AND THEY ALSO CONFIRM THE THEORETICAL RATES OF CONVERGENCE.

IN THIS WORK WE PRESENT A NEW MIXED FINITE ELEMENT METHOD FOR A CLASS OF STEADY-STATE NATURAL CONVECTION MODELS DESCRIBING THE BEHAVIOR OF NON-ISOTHERMAL INCOMPRESSIBLE FLUIDS SUBJECT TO A HEAT SOURCE. OUR APPROACH IS BASED ON THE INTRODUCTION OF A MODIFIED PSEUDOSTRESS TENSOR DEPENDING ON THE PRESSURE, AND THE DIFFUSIVE AND CONVECTIVE TERMS OF THE NAVIER?STOKES EQUATIONS FOR THE FLUID AND A VECTOR UNKNOWN INVOLVING THE TEMPERATURE, ITS GRADIENT AND THE VELOCITY. THE INTRODUCTION OF THESE FURTHER UNKNOWNS LEAD TO A MIXED FORMULATION WHERE THE AFOREMENTIONED PSEUDOSTRESS TENSOR AND VECTOR UNKNOWN, TOGETHER WITH THE VELOCITY AND THE TEMPERATURE, ARE THE MAIN UNKNOWNS OF THE SYSTEM. THEN THE ASSOCIATED GALERKIN SCHEME CAN BE DEFINED BY EMPLOYING RAVIART?THOMAS ELEMENTS OF DEGREE K FOR THE PSEUDOSTRESS TENSOR AND THE VECTOR UNKNOWN, AND DISCONTINUOUS PIECE-WISE POLYNOMIAL ELEMENTS OF DEGREE K FOR THE VELOCITY AND TEMPERATURE. WITH THIS CHOICE OF SPACES, BOTH, MOMENTUM AND THERMAL ENERGY, ARE CONSERVED IF THE EXTERNAL FORCES BELONG TO THE VELOCITY AND TEMPERATURE DISCRETE SPACES, RESPECTIVELY, WHICH CONSTITUTES ONE OF THE MAIN FEATURE OF OUR APPROACH. WE PROVE UNIQUE SOLVABILITY FOR BOTH, THE CONTINUOUS AND DISCRETE PROBLEMS AND PROVIDE THE CORRESPONDING CONVERGENCE ANALYSIS. FURTHER VARIABLES OF INTEREST, SUCH AS THE FLUID PRESSURE, THE FLUID VORTICITY, THE FLUID VELOCITY GRADIENT, AND THE HEAT-FLUX CAN BE EASILY APPROXIMATED AS A SIMPLE POSTPROCESS OF THE FINITE ELEMENT SOLUTIONS WITH THE SAME RATE OF CONVERGENCE. FINALLY, SEVERAL NUMERICAL RESULTS ILLUSTRATING THE PERFORMANCE OF THE METHOD ARE PROVIDED.

IN THIS PAPER WE CONSIDER AN AUGMENTED FULLY-MIXED VARIATIONAL FORMULATION THAT HAS BEEN RECENTLY PROPOSED FOR THE COUPLING OF THE NAVIER?STOKES EQUATIONS (WITH NONLINEAR VISCOSITY) AND THE LINEAR DARCY MODEL, AND DERIVE A RELIABLE AND EFFICIENT RESIDUAL-BASED A POSTERIORI ERROR ESTIMATOR FOR THE ASSOCIATED MIXED FINITE ELEMENT SCHEME. THE FINITE ELEMENT SUBSPACES EMPLOYED ARE PIECEWISE CONSTANTS, RAVIART?THOMAS ELEMENTS OF LOWEST ORDER, CONTINUOUS PIECEWISE LINEAR ELEMENTS, AND PIECEWISE CONSTANTS FOR THE STRAIN, CAUCHY STRESS, VELOCITY, AND VORTICITY IN THE FLUID, RESPECTIVELY, WHEREAS RAVIART?THOMAS ELEMENTS OF LOWEST ORDER FOR THE VELOCITY, PIECEWISE CONSTANTS FOR THE PRESSURE, AND CONTINUOUS PIECEWISE LINEAR ELEMENTS FOR THE TRACES, ARE CONSIDERED IN THE POROUS MEDIUM. THE PROOF OF RELIABILITY OF THE ESTIMATOR RELIES ON A GLOBAL INF?SUP CONDITION, SUITABLE HELMHOLTZ DECOMPOSITIONS IN THE FLUID AND THE POROUS MEDIUM, THE LOCAL APPROXIMATION PROPERTIES OF THE CLÉMENT AND RAVIART?THOMAS OPERATORS, AND A SMALLNESS ASSUMPTION ON THE DATA. IN TURN, INVERSE INEQUALITIES, THE LOCALIZATION TECHNIQUE BASED ON BUBBLE FUNCTIONS, AND KNOWN RESULTS FROM PREVIOUS WORKS, ARE THE MAIN TOOLS YIELDING THE EFFICIENCY ESTIMATE. FINALLY, SEVERAL NUMERICAL RESULTS CONFIRMING THE PROPERTIES OF THE ESTIMATOR AND ILLUSTRATING THE PERFORMANCE OF THE ASSOCIATED ADAPTIVE ALGORITHM ARE REPORTED.

WE HAVE RECENTLY PROPOSED A NEW FINITE ELEMENT METHOD FOR A MORE GENERAL BOUSSINESQ MODEL IN 2D GIVEN BY THE CASE IN WHICH THE VISCOSITY OF THE FLUID DEPENDS ON ITS TEMPERATURE. OUR APPROACH IS BASED ON A PSEUDOSTRESS?VELOCITY?VORTICITY MIXED FORMULATION FOR THE MOMENTUM EQUATIONS, WHICH IS SUITABLY AUGMENTED WITH GALERKIN-TYPE TERMS, COUPLED WITH THE USUAL PRIMAL FORMULATION FOR THE ENERGY EQUATION, ALONG WITH THE INTRODUCTION OF THE NORMAL HEAT FLUX ON THE BOUNDARY AS A LAGRANGE MULTIPLIER TAKING CARE OF THE FACT THAT THE PRESCRIBED TEMPERATURE THERE BECOMES AN ESSENTIAL CONDITION. THEN, FIXED-POINT ARGUMENTS USING BANACH AND BROUWER THEOREMS, IN ADDITION TO OTHER CLASSICAL TOOLS FROM FUNCTIONAL AND NUMERICAL ANALYSIS, PROVIDE SUFFICIENT CONDITIONS ENSURING WELL-POSEDNESS OF THE RESULTING CONTINUOUS AND DISCRETE SYTEMS, TOGETHER WITH THE CORRESPONDING ERROR ESTIMATES AND ASSOCIATED RATES OF CONVERGENCE. IN THE PRESENT WORK WE COMPLEMENT THESE RESULTS WITH THE DERIVATION OF A RELIABLE AND EFFICIENT RESIDUAL-BASED A POSTERIORI ERROR ESTIMATOR FOR THE AFOREMENTIONED AUGMENTED MIXED-PRIMAL FINITE ELEMENT METHOD. DUALITY TECHNIQUES, HELMHOLTZ DECOMPOSITIONS, AND THE APPROXIMATION PROPERTIES OF THE RAVIART?THOMAS AND CLÉMENT INTERPOLANTS ARE APPLIED TO OBTAIN A RELIABLE GLOBAL ERROR INDICATOR. IN TURN, STANDARD TOOLS INCLUDING THE USUAL LOCALIZATION TECHNIQUE OF BUBBLE FUNCTIONS AND INVERSE INEQUALITIES, AND A REGULARITY ASSUMPTION ORIGINALLY UTILIZED IN THE PREVIOUS WELL-POSEDNESS AND A PRIORI ERROR ANALYSES, ARE EMPLOYED TO PROVE ITS EFFICIENCY. IN BOTH CASES, RELIABILITY AND EFFICIENCY, THE ESTIMATES ARE SHOWN AT GLOBAL LEVEL. FINALLY, A RELIABLE FULLY LOCAL AND COMPUTABLE A POSTERIORI ERROR ESTIMATOR INDUCED BY THE AFOREMENTIONED ONE IS DEDUCED, AND SEVERAL NUMERICAL RESULTS ILLUSTRATING ITS PERFORMANCE AND VALIDATING THE EXPECTED BEHAVIOUR OF THE ASSOCIATED ADAPTIVE ALGORITHM ARE REPORTED.

IN THIS PAPER WE CONSIDER A PARTIALLY AUGMENTED FULLY-MIXED VARIATIONAL FORMULATION THAT HAS BEEN RECENTLY PROPOSED FOR THE COUPLING OF THE STATIONARY BRINKMAN?FORCHHEIMER AND DOUBLE DIFFUSION EQUATIONS, AND DEVELOP AN A POSTERIORI ERROR ANALYSIS FOR THE 2D AND 3D VERSIONS OF THE ASSOCIATED MIXED FINITE ELEMENT SCHEME. INDEED, WE DERIVE TWO RELIABLE AND EFFICIENT RESIDUAL-BASED A POSTERIORI ERROR ESTIMATORS FOR THIS PROBLEM ON ARBITRARY (CONVEX OR NONCONVEX) POLYGONAL AND POLYHEDRAL REGIONS. THE RELIABILITY OF THE PROPOSED ESTIMATORS DRAWS MAINLY UPON THE UNIFORM ELLIPTICITY AND INF-SUP CONDITION OF THE FORMS INVOLVED, A SUITABLE ASSUMPTION ON THE DATA, STABLE HELMHOLTZ DECOMPOSITIONS IN HILBERT AND BANACH FRAMEWORKS, AND THE LOCAL APPROXIMATION PROPERTIES OF THE CLÉMENT AND RAVIART?THOMAS OPERATORS. IN TURN, INVERSE INEQUALITIES, THE LOCALIZATION TECHNIQUE BASED ON BUBBLE FUNCTIONS, AND KNOWN RESULTS FROM PREVIOUS WORKS, ARE THE MAIN TOOLS YIELDING THE EFFICIENCY ESTIMATE. FINALLY, SEVERAL NUMERICAL EXAMPLES CONFIRMING THE THEORETICAL PROPERTIES OF THE ESTIMATORS AND ILLUSTRATING THE PERFORMANCE OF THE ASSOCIATED ADAPTIVE ALGORITHMS, ARE REPORTED. IN PARTICULAR, THE CASEOF FLOW THROUGH A 3D POROUS MEDIA WITH CHANNEL NETWORKS IS CONSIDERED.

IN THIS PAPER WE COMPLEMENT THE STUDY OF A NEW MIXED FNITE ELEMENT SCHEME, ALLOWING CONSERVATION OF MOMENTUM AND THERMAL ENERGY, FOR THE BOUSSINESQ MODEL DESCRIBING NATURAL CONVECTION AND DERIVE A RELIABLE AND EFCIENT RESIDUAL-BASED A POSTERIORI ERROR ESTIMATOR FOR THE CORRESPONDING GALERKIN SCHEME IN TWO AND THREE DIMENSIONS. MORE PRECISELY, BY EXTENDING STANDARD TECHNIQUES COMMONLY USED ON HILBERT SPACES TO THE CASE OF BANACH SPACES, SUCH AS LOCAL ESTIMATES, SUITABLE HELMHOLTZ DECOMPOSITIONS AND THE LOCAL APPROXIMATION PROPERTIES OF THE CLÉMENT AND RAVIART?THOMAS OPERATORS, WE DERIVE THE AFOREMENTIONED A POSTERIORI ERROR ESTIMATOR ON ARBITRARY (CONVEX OR NON-CONVEX) POLYGONAL AND POLYHEDRAL REGIONS. IN TURN, INVERSE INEQUALITIES, THE LOCALIZATION TECHNIQUE BASED ON BUBBLE FUNCTIONS, AND KNOWN RESULTS FROM PREVIOUS WORKS, ARE EMPLOYED TO PROVE THE LOCAL EFCIENCY OF THE PROPOSED A POSTERIORI ERROR ESTIMATOR. FINALLY, TO ILLUSTRATE THE PERFORMANCE OF THE ADAPTIVE ALGORITHM BASED ON THE PROPOSED A POSTERIORI ERROR INDICATOR AND TO CORROBORATE THE THEORETICAL RESULTS, WE PROVIDE SOME NUMERICAL EXAMPLES.

IN THIS ARTICLE, WE CONSIDER AN AUGMENTED FULLY MIXED VARIATIONAL FORMULATION THAT HAS BEEN RECENTLY PROPOSED FOR THE NONISOTHERMAL OLDROYD?STOKES PROBLEM, AND DEVELOP AN A POSTERIORI ERROR ANALYSIS FOR THE 2-D AND 3-D VERSIONS OF THE ASSOCIATED MIXED FINITE ELEMENT SCHEME. MORE PRECISELY, WE DERIVE TWO RELIABLE AND EFFICIENT RESIDUAL-BASED A POSTERIORI ERROR ESTIMATORS FOR THIS PROBLEM ON ARBITRARY (CONVEX OR NONCONVEX) POLYGONAL AND POLYHEDRAL REGIONS. THE RELIABILITY OF THE PROPOSED ESTIMATORS DRAWS MAINLY UPON THE UNIFORM ELLIPTICITY OF THE BILINEAR FORMS OF THE CONTINUOUS FORMULATION, SUITABLE ASSUMPTIONS ON THE DOMAIN AND THE DATA, STABLE HELMHOLTZ DECOMPOSITIONS, AND THE LOCAL APPROXIMATION PROPERTIES OF THE CLÉMENT AND RAVIART?THOMAS OPERATORS. ON THE OTHER HAND, INVERSE INEQUALITIES, THE LOCALIZATION TECHNIQUE BASED ON BUBBLE FUNCTIONS, AND KNOWN RESULTS FROM PREVIOUS WORKS ARE THE MAIN TOOLS YIELDING THE EFFICIENCY ESTIMATE. FINALLY, SEVERAL NUMERICAL RESULTS CONFIRMING THE PROPERTIES OF THE A POSTERIORI ERROR ESTIMATORS AND ILLUSTRATING THE PERFORMANCE OF THE ASSOCIATED ADAPTIVE ALGORITHMS ARE REPORTED.

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.

WE PROPOSE AND ANALYZE A HIGH ORDER UNFITTED MIXED FINITE ELEMENT METHOD FOR THE PSEUDOSTRESS-VELOCITY FORMULATION OF THE STOKES PROBLEM WITH DIRICHLET BOUNDARY CONDITION ON A FLUID DOMAIN WITH CURVED BOUNDARY . THE METHOD CONSISTS OF APPROXIMATING BY A POLYHEDRAL SUBDOMAIN , WITH BOUNDARY , WHERE A GALERKIN METHOD IS APPLIED TO APPROXIMATE THE SOLUTION, AND ON A TRANSFERRING TECHNIQUE, BASED ON INTEGRATING THE EXTRAPOLATED DISCRETE GRADIENT OF THE VELOCITY, TO APPROXIMATE THE DIRICHLET BOUNDARY DATA ON THE COMPUTATIONAL BOUNDARY . THE ASSOCIATED GALERKIN SCHEME IS DEFINED BY RAVIART?THOMAS ELEMENTS OF ORDER FOR THE PSEUDOSTRESS AND DISCONTINUOUS POLYNOMIALS OF DEGREE FOR THE VELOCITY. PROVIDED SUITABLE HYPOTHESES ON THE MESH NEAR THE BOUNDARY , WE PROVE WELL-POSEDNESS OF THE GALERKIN SCHEME BY MEANS OF THE BABU?KA?BREZZI THEORY AND ESTABLISH THE CORRESPONDING OPTIMAL CONVERGENCE OF . MOREOVER, FOR THE CASE WHEN IS CONSTRUCTED THROUGH A PIECEWISE LINEAR INTERPOLATION OF , WE PROPOSE A RELIABLE AND QUASI-EFFICIENT RESIDUAL-BASED A POSTERIORI ERROR ESTIMATOR. ITS DEFINITION MAKES USE OF A POSTPROCESSED VELOCITY WITH ENHANCED ACCURACY TO ACHIEVE THE SAME RATE OF CONVERGENCE OF THE METHOD WHEN THE SOLUTION IS SMOOTH ENOUGH. NUMERICAL EXPERIMENTS ILLUSTRATE THE PERFORMANCE OF THE SCHEME, SHOW THE BEHAVIOR OF THE ASSOCIATED ADAPTIVE ALGORITHM AND VALIDATE THE THEORY.

THIS PAPER DEALS WITH THE ANALYSIS OF NEW MIXED FINITE ELEMENT METHODS FOR THE BRINKMAN EQUATIONS FORMULATED IN TERMS OF VELOCITY, VORTICITY AND PRESSURE. EMPLOYING THE BABU-KA-BREZZI THEORY, IT IS PROVED THAT THE RESULTING CONTINUOUS AND DISCRETE VARIATIONAL FORMULATIONS ARE WELL-POSED. IN PARTICULAR, WE SHOW THAT RAVIART-THOMAS ELEMENTS OF ORDER K?0 FOR THE APPROXIMATION OF THE VELOCITY FIELD, PIECEWISE CONTINUOUS POLYNOMIALS OF DEGREE K+1 FOR THE VORTICITY, AND PIECEWISE POLYNOMIALS OF DEGREE K FOR THE PRESSURE, YIELD UNIQUE SOLVABILITY OF THE DISCRETE PROBLEM. ON THE OTHER HAND, WE ALSO SHOW THAT FAMILIES OF FINITE ELEMENTS BASED ON BREZZI?DOUGLAS?MARINI ELEMENTS OF ORDER K+1 FOR THE APPROXIMATION OF VELOCITY, PIECEWISE CONTINUOUS POLYNOMIALS OF DEGREE K+2 FOR THE VORTICITY, AND PIECEWISE POLYNOMIALS OF DEGREE K FOR THE PRESSURE ENSURE THE WELL-POSEDNESS OF THE ASSOCIATED GALERKIN SCHEME. WE NOTE THAT THESE FAMILIES PROVIDE EXACTLY DIVERGENCE-FREE APPROXIMATIONS OF THE VELOCITY FIELD. WE ESTABLISH A PRIORI ERROR ESTIMATES IN THE NATURAL NORMS WITH CONSTANTS INDEPENDENT OF THE VISCOSITY AND WE CARRY OUT THE RELIABILITY AND EFFICIENCY ANALYSIS OF A RESIDUAL-BASED A POSTERIORI ERROR ESTIMATOR. FINALLY, WE REPORT SEVERAL NUMERICAL EXPERIMENTS ILLUSTRATING THE BEHAVIOUR OF THE PROPOSED SCHEMES AND CONFIRMING OUR THEORETICAL RESULTS ON UNSTRUCTURED MESHES. ADDITIONAL EXAMPLES OF CASES NOT COVERED BY OUR THEORY ARE ALSO PRESENTED.