IN THIS ARTICLE, WE STUDY A MATHEMATICAL MODEL THAT REPRESENTS THE CONCENTRATION POLARIZATION AND OSMOSIS EFFECTS IN A REVERSE OSMOSIS CROSS-FLOW CHANNEL WITH DENSE MEMBRANES AT SOME OF ITS BOUNDARIES. THE FLUID IS MODELED USING THE NAVIER-STOKES EQUATIONS AND THE SOLUTION-DIFFUSION IS USED TO IMPOSE THE MOMENTUM BALANCE ON THE MEMBRANE. THE SCHEME CONSIST OF A CONFORMING FINITE ELEMENT METHOD WITH THE VELOCITY-PRESSURE FORMULATION FOR THE NAVIER-STOKES EQUATIONS, TOGETHER WITH A PRIMAL SCHEME FOR THE CONVECTION-DIFFUSION EQUATIONS. THE NITSCHE

WE PRESENT THE ANALYSIS OF AN ABSTRACT PARAMETER-DEPENDENT MIXED VARIATIONAL FORMULATION BASED ON VOLTERRA INTEGRALS OF SECOND KIND. ADAPTING THE CLASSIC MIXED THEORY IN THE VOLTERRA EQUATIONS SETTING, WE PROVE THE WELL POSEDNESS OF THE RESULTING SYSTEM. STABILITY AND ERROR ESTIMATES ARE DERIVED, WHERE ALL THE ESTIMATES ARE UNIFORM WITH RESPECT TO THE PERTURBATION PARAMETER. WE PROVIDE APPLICATIONS OF THE DEVELOPED ANALYSIS FOR A VISCOELASTIC TIMOSHENKO BEAM AND REPORT NUMERICAL TESTS FOR THIS PROBLEM. WE ALSO COMMENT, NUMERICALLY, THE PERFORMANCE OF A VISCOELASTIC REISSNER?MINDLIN PLATE.

IN THIS PAPER WE ANALYZE A POSTERIORI ERROR ESTIMATES FOR A MIXED FORMULATION OF THE LINEAR ELASTICITY EIGENVALUE PROBLEM. A POSTERIORI ESTIMATORS FOR THE NEARLY AND PERFECTLY COMPRESSIBLE ELASTICITY SPECTRAL PROBLEMS ARE PROPOSED. WITH A POST-PROCESS ARGUMENT, WE ARE ABLE TO PROVE RELIABILITY AND EFFICIENCY FOR THE PROPOSED ESTIMATORS. THE NUMERICAL METHOD IS BASED IN RAVIART-THOMAS ELEMENTS TO APPROXIMATE THE PSEUDOSTRESS AND PIECEWISE POLYNOMIALS FOR THE DISPLACEMENT. WE ILLUSTRATE OUR RESULTS WITH NUMERICAL TESTS IN TWO AND THREE DIMENSIONS.

THE AIM OF THIS PAPER IS TO ANALYZE A MIXED FORMULATION FOR THE TWO DIMENSIONAL STOKES EIGENVALUE PROBLEM WHERE THE UNKNOWNS ARE THE STRESS AND THE VELOCITY, WHEREAS THE PRESSURE CAN BE RECOVERED WITH A SIMPLE POSTPROCESS OF THE STRESS. THE STRESS TENSOR IS WRITTEN IN TERMS OF THE VORTICITY OF THE FLUID, LEADING TO AN ALTERNATIVE MIXED FORMULATION THAT INCORPORATES THIS PHYSICAL FEATURE. WE PROPOSE A MIXED NUMERICAL METHOD WHERE THE STRESS IS APPROXIMATED WITH SUITABLE NÉDELEC FINITE ELEMENTS, WHEREAS THE VELOCITY IS APPROXIMATED WITH PIECEWISE POLYNOMIALS OF DEGREE . WITH THE AID OF THE COMPACT OPERATORS THEORY WE DERIVE CONVERGENCE OF THE METHOD AND SPECTRAL CORRECTNESS. MOREOVER, WE PROPOSE A RELIABLE AND EFFICIENT A POSTERIORI ERROR ESTIMATOR FOR OUR SPECTRAL PROBLEM IN ORDER TO PROVIDE AN ADAPTIVE STRATEGY TO ACHIEVE THE OPTIMAL ORDER OF CONVERGENCE FOR NON SUFFICIENT SMOOTH EIGENFUNCTIONS. WE REPORT NUMERICAL TESTS WHERE THE SPECTRUM IS COMPUTED, TOGETHER WITH A COMPUTATIONAL ANALYSIS FOR THE PROPOSED ESTIMATOR. IN ADDITION, WE USE THE CORRESPONDING ERROR ESTIMATOR TO DRIVE AN ADAPTIVE SCHEME, AND WE REPORT THE RESULTS OF A NUMERICAL TEST, THAT ALLOW US TO ASSESS THE PERFORMANCE OF THIS APPROACH.