IN THIS PAPER WE PROPOSE AN UNCONDITIONAL ENERGY-STABLE TIME-SPLITTING FINITE-ELEMENT SCHEME FOR APPROXIMATING THE ERICKSEN--LESLIE EQUATIONS GOVERNING THE FLOW OF NEMATIC LIQUID CRYSTALS. THESE EQUATIONS ARE TO BE SOLVED FOR A VELOCITY VECTOR FIELD AND A SCALAR PRESSURE AS WELL AS A DIRECTOR VECTOR FIELD REPRESENTING THE DIRECTION ALONG WHICH THE MOLECULES OF THE LIQUID CRYSTAL ARE ORIENTED. THE ALGORITHM IS DESIGNED AT TWO LEVELS. FIRST, AT THE VARIATIONAL LEVEL, THE VELOCITY, PRESSURE, AND DIRECTOR ARE COMPUTED SEPARATELY, BUT THE DIRECTOR FIELD HAS TO BE COMPUTED TOGETHER WITH AN AUXILIARY VARIABLE (ASSOCIATED TO THE EQUILIBRIUM EQUATION FOR THE DIRECTOR) IN ORDER TO DEDUCE A PRIORI ENERGY ESTIMATES. SECOND, AT THE ALGEBRAIC LEVEL, ONE CAN AVOID COMPUTING SUCH AN AUXILIARY VARIABLE IF THIS IS APPROXIMATED BY A PIECEWISE CONSTANT FINITE-ELEMENT SPACE. THEREFORE, THESE TWO STEPS GIVE RISE TO A NUMERICAL ALGORITHM THAT COMPUTES SEPARATELY ONLY THE PRIMARY VARIABLES: VELOCITY, PRESSURE, AND DIRECTOR VECTOR. MOREOVER, WE WILL USE A PRESSURE STABILIZATION TECHNIQUE THAT ALLOWS A STABLE EQUAL-ORDER INTERPOLATION FOR THE VELOCITY AND THE PRESSURE. FINALLY, SOME NUMERICAL SIMULATIONS ARE PERFORMED IN ORDER TO SHOW THE ROBUSTNESS AND EFFICIENCY OF THE PROPOSED NUMERICAL SCHEME AND ITS ACCURACY.

WE CONSIDER GALERKIN APPROXIMATIONS FOR THE EQUATIONS MODELING THE MOTION OF AN INCOMPRESSIBLE MAGNETO-MICROPOLAR FLUID IN A BOUNDED DOMAIN. WE DERIVE AN OPTIMAL UNIFORM IN TIME ERROR BOUND IN THE H1 AND L2 -NORMS FOR THE VELOCITY. THIS IS DONE WITHOUT EXPLICIT ASSUMPTION OF EXPONENTIAL STABILITY FOR A CLASS OF SOLUTIONS CORRESPONDING TO DECAYING EXTERNAL FORCE FIELDS. OUR STUDY IS DONE FOR NO-SLIP BOUNDARY CONDITIONS, BUT THE RESULTS OBTAINED ARE EASILY EXTENDED TO THE CASE OF PERIODIC BOUNDARY CONDITIONS.

WE PROPOSE A FULLY DISCRETE SCHEME FOR APPROXIMATING A THREE-DIMENSIONAL, STRONGLY NONLINEAR MODEL OF MASS DIFFUSION, ALSO CALLED THE COMPLETE KAZHIKHOV?SMAGULOV MODEL. THE SCHEME USES A FINITE-ELEMENT APPROXIMATION FOR ALL UNKNOWNS (DENSITY, VELOCITY AND PRESSURE), EVEN THOUGH THE DENSITY LIMIT, SOLUTION OF THE CONTINUOUS PROBLEM, BELONGS TO . A FIRST-ORDER TIME DISCRETIZATION IS USED SUCH THAT, AT EACH TIME STEP, ONE ONLY NEEDS TO SOLVE TWO DECOUPLED LINEAR PROBLEMS FOR THE DISCRETE DENSITY AND THE VELOCITY?PRESSURE, SEPARATELY.