WE PROPOSE AND ANALYZE AN AUGMENTED MIXED FORMULATION FOR THE TIME-DEPENDENT BRINKMAN-FORCHHEIMER EQUATIONS WRITTEN IN TERMS OF VORTICITY, VELOCITY AND PRESSURE. THE WEAK FORMULATION IS BASED ON THE INTRODUCTION OF SUITABLE LEAST SQUARES TERMS ARISING FROM THE INCOMPRESSIBILITY CONDITION AND THE CONSTITUTIVE EQUATION RELATING THE VORTICITY AND VELOCITY. 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 BASED ON STABLE STOKES ELEMENTS FOR THE VELOCITY AND PRESSURE, AND CONTINUOUS OR DISCONTINUOUS PIECEWISE POLYNOMIAL SPACES FOR THE VORTICITY. 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 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.

IN THIS ARTICLE, WE PROPOSE A NEW FORMULATION AND A SUITABLE FINITE ELEMENT METHOD FOR THE STEADY COUPLING OF VISCOUS FLOW IN DEFORMABLE POROUS MEDIA USING DIVERGENCE-CONFORMING FILTRATION FLUXES. THE PROPOSED METHOD IS BASED ON THE USE OF PARAMETER-WEIGHTED SPACES, WHICH ALLOWS FOR A MORE ACCURATE AND ROBUST ANALYSIS OF THE CONTINUOUS AND DISCRETE PROBLEMS. FURTHERMORE, WE CONDUCT A SOLVABILITY ANALYSIS OF THE PROPOSED METHOD AND DERIVE OPTIMAL ERROR ESTIMATES IN APPROPRIATE NORMS. THESE ERROR ESTIMATES ARE SHOWN TO BE ROBUST IN A VARIETY OF REGIMES, INCLUDING THE CASE OF LARGE LAME PARAMETERS AND SMALL PERMEABILITY AND STORATIVITY COEFFICIENTS. TO ILLUSTRATE THE EFFECTIVENESS OF THE PROPOSED METHOD, WE PROVIDE A FEW REPRESENTATIVE NUMERICAL EXAMPLES, INCLUDING CONVERGENCE VERIFICATION AND TESTING OF ROBUSTNESS OF BLOCK-DIAGONAL PRECONDITIONERS WITH RESPECT TO MODEL PARAMETERS.

WE PROPOSE AND ANALYSE AN AUGMENTED MIXED FINITE ELEMENT METHOD FOR THE OSEEN EQUATIONS WRITTEN IN TERMS OF VELOCITY, VORTICITY, AND PRESSURE WITH NON-CONSTANT VISCOSITY AND HOMOGENEOUS DIRICHLET BOUNDARY CONDITION FOR THE VELOCITY. THE WEAK FORMULATION INCLUDES LEAST-SQUARES TERMS ARISING FROM THE CONSTITUTIVE EQUATION AND FROM THE INCOMPRESSIBILITY CONDITION, AND WE SHOW THAT IT SATISFIES THE HYPOTHESES OF THE BABU?KA-BREZZI THEORY. REPEATING THE ARGUMENTS OF THE CONTINUOUS ANALYSIS, THE STABILITY AND SOLVABILITY OF THE DISCRETE PROBLEM ARE ESTABLISHED. THE METHOD IS SUITED FOR ANY STOKES INF-SUP STABLE FINITE ELEMENT PAIR FOR VELOCITY AND PRESSURE, WHILE FOR VORTICITY ANY GENERIC DISCRETE SPACE (OF ARBITRARY ORDER) CAN BE USED. A PRIORI AND A POSTERIORI ERROR ESTIMATES ARE DERIVED USING TWO SPECIFIC FAMILIES OF DISCRETE SUBSPACES. FINALLY, WE PROVIDE A SET OF NUMERICAL TESTS ILLUSTRATING THE BEHAVIOUR OF THE SCHEME, VERIFYING THE THEORETICAL CONVERGENCE RATES, AND SHOWING THE PERFORMANCE OF THE ADAPTIVE ALGORITHM GUIDED BY RESIDUAL A POSTERIORI ERROR ESTIMATION.