雪花域上的局域化

Discretization of the Koch Snowflake Domain with Boundary and Interior Energies

We study the discretization of a Dirichlet form on the Koch snowflake domain and its boundary with the property that both the interior and the boundary can support positive energy. We compute eigenvalues and eigenfunctions, and demonstrate the localization of high energy eigenfunctions on the boundary via a modification of an argument of Filoche and Mayboroda. Hölder continuity and uniform approximation of eigenfunctions are also discussed.

具有分形元素的间断伽辽金方法

A discontinuous Galerkin method with fractal elements

本文提出了一种用于分形边界域的间断伽辽金有限元方法，通过分形网格精确计算局部多项式基函数，并展示了该方法在泊松方程和狄利克雷特征值问题中的有效性。

Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials

This paper concerns spectral properties of linear Schrödinger operators under oscillatory high-amplitude potentials on bounded domains. Depending on the degree of disorder, we prove the existence of spectral gaps amongst the lowermost eigenvalues and the emergence of exponentially localized states. We quantify the rate of decay in terms of geometric parameters that characterize the potential. The proofs are based on the convergence theory of iterative solvers for eigenvalue problems and their optimal local preconditioning by domain decomposition.