Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation

We continue the study of the nonconforming multiscale finite element method (Ms- FEM) introduced in 17, 14 for second order elliptic equations with highly oscillatory coefficients. The main difficulty in MsFEM, as well as other numerical upscaling methods, is the scale resonance effect. It has been show that the leading order resonance error can be effectively removed by using an over-sampling technique. Nonetheless, there is still a secondary cell resonance error of O(Є^2/h^2). Here, we introduce a Petrov-Galerkin MsFEM formulation with nonconforming multiscale trial functions and linear test functions. We show that the cell resonance error is eliminated in this formulation and hence the convergence rate is greatly improved. Moreover, we show that a similar formulation can be used to enhance the convergence of an immersed-interface finite element method for elliptic interface problems

Charged BTZ-like black hole solutions and the diffusivity-butterfly velocity relation

We show that there exists a class of charged BTZ-like black hole solutions in Lifshitz spacetime with a hyperscaling violating factor. The charged BTZ is characterized by a charge-dependent logarithmic term in the metric function. As concrete examples, we give five such charged BTZ-like black hole solutions and the standard charged BTZ metric can be regarded as a special instance of them. In order to check the recent proposed universal relations between diffusivity and the butterfly velocity, we first compute the diffusion constants of the standard charged BTZ black holes and then extend our calculation to arbitrary dimension $d$, exponents $z$ and $\theta$. Remarkably, the case $d=\theta$ and $z=2$ is a very special in that the charge diffusion $D_c$ is a constant and the energy diffusion $D_e$ might be ill-defined, but $v^2_B\tau$ diverges. We also compute the diffusion constants for the case that the DC conductivity is finite but in the absence of momentum relaxation.Comment: 30 pages, 2 figure

Hydrogen as a Source of Flux Noise in SQUIDs

Superconducting qubits are hampered by flux noise produced by surface spins from a variety of microscopic sources. Recent experiments indicated that hydrogen (H) atoms may be one of those sources. Using density functional theory calculations, we report that H atoms either embedded in, or adsorbed on, an a-Al2O3(0001) surface have sizeable spin moments ranging from 0.81 to 0.87 uB with energy barriers for spin reorientation as low as ~10 mK. Furthermore, H adatoms on the surface attract gas molecules such as O2, producing new spin sources. We propose coating the surface with graphene to eliminate H-induced surface spins and to protect the surface from other adsorbates.Comment: 12 pages, 4 figure