Mixed formulation, approximation and decoupling algorithm for a penalized nematic liquid crystals model

Abstract

A linear fully discrete mixed scheme, using C 0 finite elements in space and a semi-implicit Euler scheme in time, is considered for solving a penalized nematic liquid crystal model (of the Ginzburg-Landau type). We prove: 1) unconditional stability and convergence towards weak solutions, and 2) first-order optimal error estimates for regular solutions (but without imposing the well-known global compatibility condition for the initial pressure in the Navier-Stokes framework). These results are valid in a general connected polygon or in a Lipschitz polyhedral domain (without any constraints on its angles). Finally, since the scheme couples the unknowns, we propose several algorithms for decoupling the computation of these unknowns and establish their rates of convergence in convex domains when the mesh size is sufficiently small compared to the time step.