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Simulations of singularity dynamics in liquid crystal flows: a \(C^0\) finite element approach. (English) Zbl 1101.82039

Summary: We present a \(C^0\) finite element method for a 2D hydrodynamic liquid crystal model which is simpler than existing \(C^1\) element methods and mixed element formulation. The energy law is formally justified and the energy decay is used as a validation tool for our numerical computation. A splitting method combined with only a few fixed point iteration for the penalty term of the director field is applied to reduce the size of the stiffness matrix and to keep the stiffness matrix time-independent. The latter avoids solving a linear system at every time step and largely reduces the computational time, especially when direct linear system solvers are used. Our approach is verified by comparing its computational results with those obtained by \(C^1\) elements and by mixed formulation. Through numerical experiments of a few other splittings and explicit-implicit strategies, we recommend a fast and reliable algorithm for this model. A number of examples are computed to demonstrate the algorithm.

MSC:

82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
76A15 Liquid crystals
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76M10 Finite element methods applied to problems in fluid mechanics
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