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

Convergence of a nonconforming multiscale finite element method

The multiscale finite element method (MsFEM) [T. Y. Hou, X. H. Wu, and Z. Cai, Math. Comp., 1998, to appear; T. Y. Hou and X. H. Wu, J. Comput. Phys., 134 (1997), pp. 169-189] has been introduced to capture the large scale solutions of elliptic equations with highly oscillatory coefficients. This is accomplished by constructing the multiscale base functions from the local solutions of the elliptic operator. Our previous study reveals that the leading order error in this approach is caused by the "resonant sampling," which leads to large error when the mesh size is close to the small scale of the continuous problem. Similar difficulty also arises in numerical upscaling methods. An oversampling technique has been introduced to alleviate this difficulty [T. Y. Hou and X. H. Wu, J. Comput. Phys., 134 (1997), pp. 169-189]. A consequence of the oversampling method is that the resulting finite element method is no longer conforming. Here we give a detailed analysis of the nonconforming error. Our analysis also reveals a new cell resonance error which is caused by the mismatch between the mesh size and the wavelength of the small scale. We show that the cell resonance error is of lower order. Our numerical experiments demonstrate that the cell resonance error is generically small and is difficult to observe in practice

Effect of finite computational domain on turbulence scaling law in both physical and spectral spaces

The well-known translation between the power law of the energy spectrum and that of the correlation function or the second order structure function has been widely used in analyzing random data. Here, we show that the translation is valid only in proper scaling regimes. The regimes of valid translation are different for the correlation function and the structure function. Indeed, they do not overlap. Furthermore, in practice, the power laws exist only for a finite range of scales. We show that this finite range makes the translation inexact even in the proper scaling regime. The error depends on the scaling exponent. The current findings are applicable to data analysis in fluid turbulence and other stochastic systems

A General Theorem Relating the Bulk Topological Number to Edge States in Two-dimensional Insulators

We prove a general theorem on the relation between the bulk topological quantum number and the edge states in two dimensional insulators. It is shown that whenever there is a topological order in bulk, characterized by a non-vanishing Chern number, even if it is defined for a non-conserved quantity such as spin in the case of the spin Hall effect, one can always infer the existence of gapless edge states under certain twisted boundary conditions that allow tunneling between edges. This relation is robust against disorder and interactions, and it provides a unified topological classification of both the quantum (charge) Hall effect and the quantum spin Hall effect. In addition, it reconciles the apparent conflict between the stability of bulk topological order and the instability of gapless edge states in systems with open boundaries (as known happening in the spin Hall case). The consequences of time reversal invariance for bulk topological order and edge state dynamics are further studied in the present framework.Comment: A mistake corrected in reference

On the application of the depth-averaged random walk method to solute transport simulations

Most numerical studies on the solute transport problems relies on mesh-based methods, and complicated schemes have been developed to enhance numerical stability and reduce artificial diffusion. This paper systematically studies the depth-averaged random walk scheme, which is a meshfree method with the merits of being highly robust and free of numerical diffusion. Firstly, the model is used to solve an instantaneous release problem in uniform flows. Extensive parametric studies are carried out to investigate the influences of the number of particles and the size of time steps. The predictions are found to be independent of time steps but are sensitive to the particle numbers. Secondly, the model is applied to the solute transport along a tidal estuary subject to extensive wetting and drying during tidal oscillations. Finally, the model is applied to investigate the wind-induced chaotic mixing in a shallow basin. The effect of diffusion on the chaotic mixing is investigated. This study proposes a generic sampling method to interpret the output of the random walk method and highlights the importance of accurately taking diffusion into account in studying the mixing phenomena. The sampling technique also offers a guideline for estimating the total number of particles needed in the application.Royal Academy of Engineering UK-China Urban Flooding Research Impact Programme (UUFRIP\100051), the 111 Project (B17015), China Scholarship Counci

High sensitivity microwave detection using a magnetic tunnel junction in the absence of an external applied magnetic field

In the absence of any external applied magnetic field, we have found that a magnetic tunnel junction (MTJ) can produce a significant output direct voltage under microwave radiation at frequencies, which are far from the ferromagnetic resonance condition, and this voltage signal can be increase by at least an order of magnitude by applying a direct current bias. The enhancement of the microwave detection can be explained by the nonlinear resistance/conductance of the MTJs. Our estimation suggests that optimized MTJs should achieve sensitivities for non-resonant broadband microwave detection of about 5,000 mV/mW

Error Performance of Double Space Time Transmit Diversity Systems

The theoretical error performance of double space time transmit diversity (DSTTD) system with optimum combining receiver is analyzed in this paper. by employing both spatial multiplexing and transmit diversity in one system, DSTTD provides practical tradeoff between system spectral efficiency and diversity gain. We derive exact analytical expressions to describe the symbol error rate for DSTTD systems. The effects of both diversity gain and antenna interference introduced by spatial multiplexing are quantified in the results. In addition, the performance of DSTTD system with successive interference cancellation is also investigated. Simulation results are in excellent agreement with the theoretical results obtained in this paper

Changes of anti-glucosidase content and some other characteristics in mulberry juice during fermentation with Leuconostoc mesenteroides

Changes in microbial population, pH, sugar, organic acid, anthocyanins, total soluble phenolics, and anti-glucosidase contents were measured during fermentation of mulberry juice at 30 °C by probiotic Leuconostoc mesenteroides showing rapid growth after an approximately 1-day lag phase and reaching a maximum of 8.6 log CFU ml−1 after 4 d. During the rapid growth phase, the main mulberry juice sugars, glucose and fructose, were largely consumed, and the acidic metabolites, lactic acid and acetic acid, were produced accordingly. A slow decrease in the concentration of the main organic acid, citric acid, was also observed during fermentation. After 4 d fermentation, anthocyanin content showed a 44.4% reduction, but the total amount of soluble phenolics and α-glucosidase inhibitory activity showed no significant changes (P>0.05). This suggests that L. mesenteroides fermentation of mulberry juice is a good strategy to enhance its probiotic value and to decrease the sugar content without changing the anti-glucosidase activity, which is required to reduce postprandial rise in blood glucose

The Degasperis-Procesi equation with self-consistent sources

The Degasperis-Procesi equation with self-consistent sources(DPESCS) is derived. The Lax representation and the conservation laws for DPESCS are constructed. The peakon solution of DPESCS is obtained.Comment: 15 page