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

Application of synthetic polymers as adsorbents for the removal of cadmium from aqueous solutions: batch experimental studies

In the present study seven synthetic polymers were used as adsorbents for the removal of Cd(II) from aqueous solution. The equilibrium studies were systematically carried out in a batch process, covering various process parameters that include agitation time, adsorbent dosage, and pH of the aqueous solution.The analyzing system was an atomic absorption spectrometer (Perkin-Analyst 100). It was observed in adsorption and desorption tests that synthetic polymers showed significant pH dependence, which affected the removal efficiency, robustly. Adsorption behavior was found to follow Freundlich and Longmuir isotherms. A regeneration study was also carried out

A model for two-component aerosol coagulation and phase separation: A method for changing the growth rate of nanoparticles, Chem

Abstract In previous studies of particle growth we have synthesized binary metal oxide aerosols and have observed the evolution of internal phase segregation during growth of molten nanodroplets. We have also generated NaCl=metal aerosols in which the metal nanoparticle is enveloped within a salt droplet. In both systems the nanoparticles were grown in the molten state. In this paper we propose a model which incorporates phase segregation in a binary aerosol. The model assumes that complete phase segregation is the thermodynamically favored state, that no thermodynamic activation energy exists, and that the phase segregation process is kinetically controlled. The results indicate that a steady state behavior can be reached in which the characteristic time for aerosol coagulation and the characteristic time for the growth of the minority phase coincide such that the number of distinct segregated entities within each aerosol droplet is constant. The results suggest what we believe to be an important concept that can be utilized in materials synthesis. This is that the major phase can be used to moderate the growth rate of the minor phase by changing the characteristic encounter frequency and therefore the eventual growth rate of the minority phase. In particular, temperature, which does not play an important role in aerosol coagulation, is seen to be a very sensitive parameter for the growth of the minority phase nanoparticles. We discuss the parameter space necessary for this to occur.

Variable Domain Eigenvalue Problems for the Laplace Operator with Density

Abstract: We consider variable domain eigenvalue problems for the placeLaplace operator with density function. The first variation of the eigenvalues is calculated with respect to domain, their properties are investigated when the domain varies

Linearly constrained evolutions of critical points and an application to cohesive fractures

We introduce a novel constructive approach to define time evolution of critical points of an energy functional. Our procedure, which is different from other more established approaches based on viscosity approximations in infinite dimension, is prone to efficient and consistent numerical implementations, and allows for an existence proof under very general assumptions. We consider in particular rather nonsmooth and nonconvex energy functionals, provided the domain of the energy is finite dimensional. Nevertheless, in the infinite dimensional case study of a cohesive fracture model, we prove a consistency theorem of a discrete-to-continuum limit. We show that a quasistatic evolution can be indeed recovered as a limit of evolutions of critical points of finite dimensional discretizations of the energy, constructed according to our scheme. To illustrate the results, we provide several numerical experiments both in one and two dimensions. These agree with the crack initiation criterion, which states that a fracture appears only when the stress overcomes a certain threshold, depending on the material