NUMERICAL ANALYSIS OF A GRAD-DIV STABILIZATION METHOD FOR THE OLDROYD MODEL OF ORDER ONE

This paper analyzes an inf-sup stable Galerkin mixed finite element method with a grad-div stabilization for the equation of the motion of the fluid arising in the Oldroyd model of order one. The main idea of the grad-div stabilization method is to add a stabilization term to the Galerkin approximation, which is very effective at a high Reynolds number. Optimal error bounds for the velocity in $ L^\infty( {\bf L}^2) $-norm and the pressure in $ L^2(L^2) $-norm are derived in the semidiscrete case with time remaining continuous. Then, a fully discrete scheme is analyzed by employing the backward Euler method, and optimal error estimates are derived. All these estimates are obtained with constants independent of the inverse of viscosity and for both the cases when the solution is as smooth as we want (has to satisfy nonlocal compatibility conditions) and when the solution is just smooth (compatibility conditions are no longer needed). Finally, we present some numerical results in support of our theoretical findings.