We consider the numerical approximations of the classical phase-field vesicle membrane models proposed a decade ago in [450, 198 (2004), J. Comput. Phys.]. We first reformulate the model derived from an energetic variational formulation into a form which is suitable for numerical approximation, and establish the energy dissipation law. Then, we develop a stabilized, decoupled, time discretization scheme for the coupled nonlinear system. The scheme is unconditionally energy stable and leads to linear and decoupled elliptic equations to be solved at each time step. Stability analysis and ample numerical simulations are presented thereafter.
In this paper we define a Rankin-Selberg L-function attached to automorphic cuspidal representations of GLm(A E ) × GL m (A F ) over cyclic algebraic number fields E and F which are invariant under the Galois action, by exploiting a result proved by Arthur and Clozel, and prove a prime number theorem for this L-function.
Study of phase transition in homogeneous,rigid extended nematics and magnetic suspensions using an order-reduction method Zhou, Hong Study of phase transition in homogeneous, rigid extended nematics and magnetic suspensions using an order-reduction method (with G. Ji, Q. Wang and P. Zhang), Physics of Fluids, 18, 123103, 2006. We study the phase transition in rigid extended nematics and magnetic suspensions by solving the Smoluchowski equation for magnetically polarized rigid nematic polymers and suspensions in equilibrium, in which the molecular interaction is modeled by a dipolar and excluded volume potential. The equilibrium solution ͑or the probability distribution of the molecular distribution͒ is given by a Boltzmann distribution parametrized by the ͑first-order͒ polarity vector and the ͑second-order͒ nematic order tensor along with material parameters. We show that the polarity vector coincides with one of the principal axes of the nematic order tensor so that the equilibrium distribution can be reduced to a Boltzmann distribution parametrized by three scalar order parameters, i.e., a polar order parameter and two nematic order parameters, governed by three nonlinear algebraic-integral equations. This reduction in the degree of freedom from 8 ͑3 in the polarity vector and 5 in the nematic order tensor͒ to 3 significantly simplifies the solution procedure and allows one to conduct a complete analysis on bifurcation diagrams of the order parameters with respect to the material parameters. The stability of the equilibria is inferred from the second variation of the free energy density.
Abstract. We derive sharp L ∞ (L 1 ) a posteriori error estimates for initial boundary value problems of nonlinear convection-diffusion equations of the form ∂u ∂tunder the nondegeneracy assumption A (s) > 0 for any s ∈ R. The problem displays both parabolic and hyperbolic behavior in a way that depends on the solution itself. It is discretized implicitly in time via the method of characteristic and in space via continuous piecewise linear finite elements. The analysis is based on the Kružkov "doubling of variables" device and the recently introduced "boundary layer sequence" technique to derive the entropy error inequality on bounded domains. The derived a posteriori error estimators have the correct convergence order in the region where the solution is smooth and recover the standard a posteriori error estimators known for parabolic equations with strong diffusions.
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