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The universality for the eigenvalue spacing statistics of generalized Wigner matrices was established in our previous work \cite{EYY} under certain conditions on the probability distributions of the matrix elements. A major class of probability measures excluded in \cite{EYY} are the Bernoulli measures. In this paper, we extend the universality result of \cite{EYY} to include the Bernoulli measures so that the only restrictions on the probability distributions of the matrix elements are the subexponential decay and the normalization condition that the variances in each row sum up to one. The new ingredient is a strong local semicircle law which improves the error estimate on the Stieltjes transform of the empirical measure of the eigenvalues from the order $(N \eta)^{-1/2}$ to $(N \eta)^{-1}$. Here $\eta$ is the imaginary part of the spectral parameter in the definition of the Stieltjes transform and $N$ is the size of the matrix.

In this paper, we propose and analyze a mixed $H^1$-conforming finite element method for solving Maxwell's equations in terms of electric field and Lagrange multiplier, where the multiplier is introduced accounting for the divergence constraint. We mainly focus on the case that the physical domain is non-convex and its boundary includes reentrant corners or edges, which may lead the solution of Maxwell's equations to be a non-$H^1$ very weak function and thus causes many numerical difficulties. The proposed method is formulated in the stabilized form by adding an additional mesh-dependent stabilization term to the mixed variational formulation. A general framework of stability and error analysis is established. Specifically, a pair of $H^1$-conforming finite elements, namely the $CP_2$-$P_1$ elements, for electric field and multiplier is studied and its stability and error bounds are also derived. Numerical experiments for source problems as well as eigenvalue problems on the $L$-shaped and cracked domains are presented to illustrate the high performance of the proposed mixed $H^1$-conforming finite element method.

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No abstract uploaded!