An energy-stable convex splitting for the phase-field crystal equation

The phase-field crystal equation is solved using a finite element discretization.A mass-conserving, energy-stable, second-order time discretization is developed.The results are proved rigorously, and verified numerically.The implementation is done in PetIGA, an open source isogeometric analysis framework.Three dimensional results showcase the robustness of the method. The phase-field crystal equation, a parabolic, sixth-order and nonlinear partial differential equation, has generated considerable interest as a possible solution to problems arising in molecular dynamics. Nonetheless, solving this equation is not a trivial task, as energy dissipation and mass conservation need to be verified for the numerical solution to be valid. This work addresses these issues, and proposes a novel algorithm that guarantees mass conservation, unconditional energy stability and second-order accuracy in time. Numerical results validating our proofs are presented, and two and three dimensional simulations involving crystal growth are shown, highlighting the robustness of the method.

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