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We prove the Poisson geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry), which is also a generalization of Conn’s linearization theorem.

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In this follow-up work, we extend the discontinuous Galerkin (DG) methods previously developed on rectangular meshes \cite{previous} to triangular meshes. The DG schemes in \cite{previous} are both optimally convergent and energy conserving. However, as we shall see in the numerical results section, the DG schemes on triangular meshes only have suboptimal convergence rate. We prove the energy conservation and an error estimate for the semi-discrete schemes. The stability of the fully discrete scheme is proved and its error estimate is stated. We present extensive numerical results with convergence consistent of our error estimate, and simulations of wave propagation in Drude metamaterials to demonstrate the flexibility of triangular meshes.

We present a generalization of the Bogomolov-Miyaoka-Yau inequality to Deligne-Mumford surfaces of general type.