Asymptotic behavior of solutions to an area-preserving motion by crystalline curvature

summary:Asymptotic behavior of solutions of an area-preserving crystalline curvature flow equation is investigated. In this equation, the area enclosed by the solution polygon is preserved, while its total interfacial crystalline energy keeps on decreasing. In the case where the initial polygon is essentially admissible and convex, if the maximal existence time is finite, then vanishing edges are essentially admissible edges. This is a contrast to the case where the initial polygon is admissible and convex: a solution polygon converges to the boundary of the Wulff shape without vanishing edges as time tends to infinity

Laplacian regularized eikonal equation with Soner boundary condition on polyhedral meshes

In this paper, we propose a numerical algorithm based on a cell-centered finite volume method to compute a distance from given objects on a three-dimensional computational domain discretized by polyhedral cells. Inspired by the vanishing viscosity method, a Laplacian regularized eikonal equation is solved and the Soner boundary condition is applied to the boundary of the domain to avoid a non-viscosity solution. As the regularization parameter depending on a characteristic length of the discretized domain is reduced, a corresponding numerical solution is calculated. A convergence to the viscosity solution is verified numerically as the characteristic length becomes smaller and the regularization parameter accordingly becomes smaller. From the numerical experiments, the second experimental order of convergence in the $L^1$ norm error is confirmed for smooth solutions. Compared to solve a time-dependent form of eikonal equation, the Laplacian regularized eikonal equation has the advantage of reducing computational cost dramatically when a more significant number of cells is used or a region of interest is far away from the given objects. Moreover, the implementation of parallel computing using domain decomposition with $1$-ring face neighborhood structure can be done straightforwardly by a standard cell-centered finite volume code

Front matter

summary:The Equadiff is a series of biannual conferences on mathematical analysis, numerical approximation and applications of differential equations. Proceedings of Equadiff 2017 Conference contain peer-reviewed contributions of participants of the conference. The proceedings cover a wide range of topics presented by plenary, minisymposia and contributed talks speakers. The scope of papers ranges from ordinary differential equations, differential inclusions and dynamical systems towards qualitative and numerical analysis of partial differential equations, stochastic PDEs and their applications