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DCDS

We present unconditionally stable and convergent numerical sche- mes for gradient flows with energy of the form $ \int_\Omega( F(\nabla\phi(\x)) + \frac{\epsilon^2}{2}|\Delta\phi(\x)|^2 )$dx. The construction of the schemes involves an appropriate extension of Eyre's idea of convex-concave decomposition of the energy functional. As an application, we derive unconditionally stable and convergent schemes for epitaxial film growth models with slope selection (F(y)= 1/4(|y|

^{2}-1)^{2}) and without slope selection (F(y)= -1/2ln(1+|y|^{2})). We conclude the paper with some preliminary computations that employ the proposed schemes.
DCDS-B

In this paper we devise and analyze a mixed discontinuous Galerkin finite element method for a modified Cahn-Hilliard equation that models phase separation in diblock copolymer melts. The time discretization is based on a convex splitting of the energy of the equation. We prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system, unconditionally uniquely solvable, and convergent in the natural energy norm with optimal rates. We describe an efficient nonlinear multigrid solver for advancing our semi-implicit scheme in time and conclude the paper with numerical tests confirming the predicted convergence rates and suggesting the near-optimal complexity of the solver.

DCDS-B

In this paper, we prove the existence and uniqueness of a Gevrey regularity solution for a class of nonlinear bistable gradient flows, where with the energy may be decomposed into purely convex and concave parts. Example equations include certain epitaxial thin film growth models and phase field crystal models. The energy dissipation law implies a bound in the leading Sobolev norm. The polynomial structure of the nonlinear terms in the chemical potential enables us to derive a local-in-time solution with Gevrey regularity, with the existence time interval length dependent on a certain $H^m$ norm of the initial data. A detailed Sobolev estimate for the gradient equations results in a uniform-in-time-bound of that $H^m$ norm, which in turn establishes the existence of a global-in-time solution with Gevrey regularity.

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