GAMMA-CONVERGENT PROJECTION-FREE FINITE ELEMENT METHODS FOR NEMATIC LIQUID CRYSTALS: THE ERICKSEN MODEL

The Ericksen model for nematic liquid crystals couples a director field with a scalar degree of orientation variable and allows the formation of various defects with finite energy. We propose a simple but novel finite element approximation of the problem that can be implemented easily within standard finite element packages. Our scheme is projection-free and thus circumvents the use of weakly acute meshes, which are quite restrictive in three dimensions but are required by recent algorithms for convergence. We prove stability and Γ-convergence properties of the new method in the presence of defects. We also design an effective nested gradient flow algorithm for computing minimizers that controls the violation of the unit-length constraint of the director. We present several simulations in two and three dimensions that document the performance of the proposed scheme and its ability to capture quite intriguing defects

Equilibrium analysis of an immersed rigid leaflet by the virtual element method

We study, both theoretically and numerically, the equilibrium of a hinged rigid leaflet with an attached rotational spring, immersed in a stationary incompressible fluid within a rigid channel. Through a careful investigation of the properties of the domain functional describing the angular momentum exerted by the fluid on the leaflet (which depends on both the leaflet angular position and its thickness), we identify sufficient conditions on the spring stiffness function for the existence (and uniqueness) of equilibrium positions. This study resorts to techniques from shape differential calculus. We propose a numerical technique that exploits the mesh flexibility of the Virtual Element Method (VEM). A (polygonal) computational mesh is generated by cutting a fixed background grid with the leaflet geometry, and the problem is then solved with stable VEM Stokes elements of degrees 1 and 2 combined with a bisection algorithm. We prove quasi-optimal error estimates and present a large array of numerical experiments to document the accuracy and robustness with respect to degenerate geometry of the proposed methodology

Qualitative behavior of solutions for thermodynamically consistent Stefan problems with surface tension

The qualitative behavior of a thermodynamically consistent two-phase Stefan problem with surface tension and with or without kinetic undercooling is studied. It is shown that these problems generate local semiflows in well-defined state manifolds. If a solution does not exhibit singularities in a sense made precise below, it is proved that it exists globally in time and its orbit is relatively compact. In addition, stability and instability of equilibria is studied. In particular, it is shown that multiple spheres of the same radius are unstable, reminiscent of the onset of Ostwald ripening.Comment: 56 pages. Expanded introduction, added references. This revised version is published in Arch. Ration. Mech. Anal. (207) (2013), 611-66