Multiphase Mean Curvature Flow: Uniqueness Properties of Weak Solution Concepts and Phase-Field Approximations

 Topology changes occur naturally in geometric evolution equations like mean curvature flow. As classical solution concepts break down at such geometric singularities, the use of weak solution concepts becomes necessary in order to describe topological changes.
For two-phase mean curvature flow, the theory of viscosity solutions by Chen-Giga-Goto and Evans-Spruck provides a concept of weak solutions with basically optimal existence and uniqueness properties.

Approximation of fractional operators and fractional PDEs using a sinc-basis

We introduce a spectral method to approximate PDEs involving the fractional Laplacian with zero exterior condition. Our approach is based on interpolation by tensor products of sinc-functions, which combine a simple representation in Fourier-space with fast enough decay to suitably approximate the bounded support of solutions to the Dirichlet problem. This yields a numerical complexity of O(NlogN) for the application of the operator to a discretization with N degrees of freedom.

Constructing robust high order entropy stable discontinuous Galerkin methods

High order methods are known to be unstable when applied to nonlinear conservation laws whose solutions exhibit shocks and turbulence. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi-discrete entropy inequality independently of numerical resolution or solution regularization and shock capturing.