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 DEVELOP AN A POSTERIORI ERROR ANALYSIS OF A NEW MOMENTUM CONSERVATIVE MIXED FINITE ELEMENT METHOD RECENTLY INTRODUCED FOR THE STEADY-STATE NAVIER?STOKES PROBLEM 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, AND SUITABLE HELMHOLTZ DECOMPOSITIONS, WE DERIVE A RELIABLE AND EFFICIENT RESIDUAL-BASED A POSTERIORI ERROR ESTIMATOR FOR THE CORRESPONDING MIXED FINITE ELEMENT SCHEME ON ARBITRARY (CONVEX OR NON-CONVEX) POLYGONAL AND POLYHEDRAL REGIONS. ON THE OTHER HAND, INVERSE INEQUALITIES, THE LOCALIZATION TECHNIQUE BASED ON BUBBLE FUNCTIONS, AMONG OTHER TOOLS, ARE EMPLOYED TO PROVE THE EFFICIENCY OF THE PROPOSED A POSTERIORI ERROR INDICATOR. FINALLY, SEVERAL NUMERICAL RESULTS CONFIRMING THE PROPERTIES OF THE ESTIMATOR AND ILLUSTRATING THE PERFORMANCE OF THE ASSOCIATED ADAPTIVE ALGORITHM ARE REPORTED.

WE PROPOSE AND ANALYZE A NEW MIXED FORMULATION FOR THE BRINKMAN-FORCHHEIMER EQUATIONS FOR UNSTEADY FLOWS. BESIDES THE VELOCITY, OUR APPROACH INTRODUCES THE VELOCITY GRADIENT AND A PSEUDOSTRESS TENSOR AS FURTHER UNKNOWNS. AS A CONSEQUENCE, WE OBTAIN A THREE-FIELD BANACH SPACES-BASED MIXED VARIATIONAL FORMULATION, WHERE THE AFOREMENTIONED VARIABLES ARE THE MAIN UNKNOWNS OF THE SYSTEM. WE ESTABLISH EXISTENCE AND UNIQUENESS OF A SOLUTION TO THE WEAK FORMULATION, AND DERIVE THE CORRESPONDING STABILITY BOUNDS, EMPLOYING CLASSICAL RESULTS ON NONLINEAR MONOTONE OPERATORS. WE THEN PROPOSE A SEMIDISCRETE CONTINUOUS-IN-TIME APPROXIMATION ON SIMPLICIAL GRIDS BASED ON THE RAVIART-THOMAS ELEMENTS OF DEGREE K >= 0 FOR THE PSEUDOSTRESS TENSOR AND DISCONTINUOUS PIECEWISE POLYNOMIALS OF DEGREE K FOR THE VELOCITY AND THE VELOCITY GRADIENT. IN ADDITION, BY MEANS OF THE BACKWARD EULER TIME DISCRETIZATION, WE INTRODUCE A FULLY DISCRETE FINITE ELEMENT SCHEME. WE PROVE WELL-POSEDNESS AND DERIVE THE STABILITY BOUNDS FOR BOTH SCHEMES, AND UNDER A QUASI-UNIFORMITY ASSUMPTION ON THE MESH, WE ESTABLISH THE CORRESPONDING ERROR ESTIMATES. WE PROVIDE SEVERAL NUMERICAL RESULTS VERIFYING THE THEORETICAL RATES OF CONVERGENCE AND ILLUSTRATING THE PERFORMANCE AND FLEXIBILITY OF THE METHOD FOR A RANGE OF DOMAIN CONFIGURATIONS AND MODEL PARAMETERS. (C) 2022 ELSEVIER B.V. ALL RIGHTS RESERVED.