Keywords: Complex Fluids, Complactness Abstract: A classical result of P. Lax states that ``a (linear) numerical scheme converges if and only if it is stable and consist''. For nonlinear problems this statement needs to augmented to include a compactness hypotheses sufficient to guarantee convergence of the nonlinear terms. This talk will focus on the development of numerical schemes for parabolic equations that are stable and inherit compactness properties of the underlying partial differential equations. I will present a discrete analog of the classical Lions-Aubin compactness theorem and use it to establish convergence of numerical schemes for fluids transporting membranes, and the Ericksen Leslie equations for (nematic) liquid crystals. The talk will finish with some open problems that arise in the numerical simulation of this class of problems.