Uniform convergence of Galerkin‐multigrid methods for nonconforming finite elements for second‐order problems with less than full elliptic regularity

In this article we prove uniform convergence estimates for the recently developed Galerkin‐multigrid methods for nonconforming finite elements for second‐order problems with less than full elliptic regularity. These multigrid methods are defined in terms of the “Galerkin approach,” where quadratic forms over coarse grids are constructed using the quadratic form on the finest grid and iterated coarse‐to‐fine intergrid transfer operators. Previously, uniform estimates were obtained for problems with full elliptic regularity, whereas these estimates are derived with less than full elliptic regularity here. Applications to the nonconforming P₁, rotated Q₁, and Wilson finite elements are analyzed. The result applies to the mixed method based on finite elements that are equivalent to these nonconforming elements. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 203–217, 2002; DOI 10.1002/num.10004     

Micro-/Mesoporous conjugated polymer based on star-shaped triazine-functional triphenylamine framework as the performance-improved cathode of Li-organic battery

A novel organic conjugated polymer based on star-shaped triazine-functional triphenylamine framework poly[1,3,5-tris(4-diphenylamino-phenyl)triazine] (PTDAPTz) is designed and synthesized successfully by FeCl₃-catalysted chemical oxidative polymerization. The polymer PTDAPTz powder exhibits a compactly packed pleated skirt shape-like morphology with a high surface area (~930 m² g⁻¹) and a bimodal pore size distribution ranging from micropores (~0.55 nm) to small diameter mesopores (~2–6 nm). As explored as the cathode material, the obtained PTDAPTz presents the double charge–discharge process characteristics of both the free radical redox of triphenylamine unit and the bipolar redox of triazine unit in the polymer and a well-defined multistage charge/discharge voltage plateau (~3.8 V for p-doped and ~2.0 V for n-doped) during the charge–discharge process. Also, the PTDAPTz demonstrates an improved capacity (stabilized at 123 mA h g⁻¹ until 50th cycle) and the enhanced rate performance compared to polytriphenylamine (PTPAn). Specially, the discharge curve for the part of triphenylamine unit presents an obviously improved discharge plateau (~3.8 V for PTDAPTz compared to ~3.6 V for PTPAn) due to the electron-withdrawing effect of the triazine unit to triphenylamine. The elaborate structural design and created micro-/mesoporous morphology with the double charge–discharge process make PTDAPTz a potential candidate as the performance-improved cathode of Li-organic battery. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2018 , 56, 2574–2583     

Water-Mediated Selective Synthesis of Pyrazolo[1,5-a]quinazolin-5(4H)-ones and [1,2,4]Triazolo[1,5-a]quinazolin-5(4H)-one via Copper-Catalyzed Cascade Reactions

A convenient and sustainable synthesis of pyrazolo[1,5-a]quinazolin-5(4H)-ones and [1,2,4]triazolo[1,5-a]quinazolin-5(4H)-one through copper-catalyzed cascade reactions of 2-bromobenzoates with 1H-pyrazol-5-amines or 1H-1,2,4-triazol-5-amine under ligand-free conditions in water is presented. It is notable that aqueous medium turned out to be crucial for the chemoselective formation of the title compounds. Compared with literature protocols, this new method showed advantages such as simple and sustainable procedure, commercially available starting materials, and convenient reuse of the reaction medium together with the copper catalyst.     

Numerical Experiments with Various Formulations for Two Phase Flow in Petroleum Reservoirs*

Two phase immiscible flow in petroleum reservoirs is considered. Various formulations of the governing equations that describe this flow, including phase, global, and weighted formulations, are numerically experimented. Mixed finite element methods are used to solve these formulations. Our experiments show that the numerical results obtained using the phase and global formulations match well in terms of production rates, characterization curves, and water cuts.     

Multiscale methods for elliptic homogenization problems

In this article we study two families of multiscale methods for numerically solving elliptic homogenization problems. The recently developed multiscale finite element method [Hou and Wu, J Comp Phys 134 (1997), 169–189] captures the effect of microscales on macroscales through modification of finite element basis functions. Here we reformulate this method that captures the same effect through modification of bilinear forms in the finite element formulation. This new formulation is a general approach that can handle a large variety of differential problems and numerical methods. It can be easily extended to nonlinear problems and mixed finite element methods, for example. The latter extension is carried out in this article. The recently introduced heterogeneous multiscale method [Engquist and Engquist, Comm Math Sci 1 (2003), 87–132] is designed for efficient numerical solution of problems with multiscales and multiphysics. In the second part of this article, we study this method in mixed form (we call it the mixed heterogeneous multiscale method). We present a detailed analysis for stability and convergence of this new method. Estimates are obtained for the error between the homogenized and numerical multiscale solutions. Strategies for retrieving the microstructural information from the numerical solution are provided and analyzed. Relationship between the multiscale finite element and heterogeneous multiscale methods is discussed. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Comparison of GMRES and ORTHOMIN for the black oil model on unstructured grids

This paper addresses an application of ORTHOMIN and GMRES to petroleum reservoir simulation using the black oil model on unstructured grids. Comparisons between these two algorithms are presented in terms of storage and total flops per restart step. Numerical results indicate that GMRES is faster than ORTHOMIN for all tested petroleum reservoir problems, particularly for large scale problems. The control volume function approximation method is utilized in the discretization of the governing equations of the black oil model. This method can accurately approximate both the pressure and velocity in the simulation of multiphase flow in porous media, effectively reduce grid orientation effects, and be easily applied to arbitrarily shaped control volumes. It is particularly suitable for hybrid grid reservoir simulation. Copyright © 2006 John Wiley & Sons, Ltd.     

Degenerate two-phase incompressible flow II: regularity, stability and stabilization

In this paper, we analyze a coupled system of highly degenerate elliptic-parabolic partial differential equations for two-phase incompressible flow in porous media. This system involves a saturation and a global pressure (or a total flow velocity). First, we show that the saturation is Hölder continuous both in space and time and the total velocity is Hölder continuous in space (uniformly in time). Applying this regularity result, we then establish the stability of the saturation and pressure with respect to initial and boundary data, from which uniqueness of the solution to the system follows. Finally, we establish a stabilization result on the asymptotic behavior of the saturation and pressure; we prove that the solution to the present system converges (in appropriate norms) to the solution of a stationary system as time goes to infinity. An example is given to show typical regularity of the saturation.     

Numerical study of the HP version of mixed discontinuous finite element methods for reaction‐diffusion problems: The 1D case

This article studies the stability and convergence of the hp version of the three families of mixed discontinuous finite element (MDFE) methods for the numerical solution of reaction‐diffusion problems. The focus of this article is on these problems for one space dimension. Error estimates are obtained explicitly in the grid size h, the polynomial degree p, and the solution regularity; arbitrary space grids and polynomial degree are allowed. These estimates are asymptotically optimal in both h and p for some of these methods. Extensive numerical results to show convergence rates in h and p of the MDFE methods are presented. Theoretical and numerical comparisons between the three families of MDFE methods are described. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 525–553, 2003     

In-situ preparation and electrochromic properties of TiO2/PTPA-HTAN core-shell nanocomposite film

A new star-shaped structure conjugated microporous polymers, poly (2,8,14-tri[4-diphenyl-benzene]-hexaazatrinaphthylene) (PTPA-HATN), was designed and in-situ electrochemically polymerized on the surfaces of FTO electrodes with a directional alignment TiO₂ nanorod array to obtain TiO₂/PTPA-HATN core-shell nanocomposite films. Compared with the PTPA-HATN film, the TiO₂/PTPA-HATN composite film exhibits higher optical contrast and faster response time, with contrast of 57% at 783 nm, coloring time of 3.62 s and discoloring time of 2.55 s (43%, 4.63 s and 4.77 s for PTPA-HATN film, respectively). After 400 cycles, the contrast of nanocomposite film decreased by 28%, while the PTPA-HATN film basically lost its electrochromic properties. A simple three-layer EC prototype device based on TiO₂/PTPA-HATN nanocomposite film constructed with hydrogel electrolyte clearly shows color changes at different voltages. On the one hand, the formation of core-shell porous nanostructure of TiO₂/PTPA-HATN composite film provides a larger ion doping/de-doping interface, shortening the average diffusion length of ions. On the other hand, the large indented polymer-nanorods contact interface makes it difficult for the polymer to detach from the electrode, thus significantly improving the cyclic stability of the composite film.