Weak Derivatives and Poincaré–Sobolev Estimates

2015 ◽  
pp. 479-518
Keyword(s):  
Sobolev Estimates ◽  
Weak Derivatives

A New Weak Galerkin Finite Element Scheme for the Brinkman Model

2016 ◽  
Vol 19 (5) ◽  
pp. 1409-1434 ◽  
Author(s):  
Qilong Zhai ◽  
Ran Zhang ◽  
Lin Mu
Keyword(s):  
Finite Element ◽  
Three Dimensional ◽  
Optimal Order ◽  
Brinkman Model ◽  
Weak Galerkin ◽  
Order Error ◽  
Brinkman Equations ◽  
Generalized Distributions ◽  
Variational Form ◽  
Weak Derivatives

AbstractThe Brinkman model describes flow of fluid in complex porous media with a high-contrast permeability coefficient such that the flow is dominated by Darcy in some regions and by Stokes in others. A weak Galerkin (WG) finite element method for solving the Brinkman equations in two or three dimensional spaces by using polynomials is developed and analyzed. The WG method is designed by using the generalized functions and their weak derivatives which are defined as generalized distributions. The variational form we considered in this paper is based on two gradient operators which is different from the usual gradient-divergence operators for Brinkman equations. The WG method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity. Optimal-order error estimates are established for the corresponding WG finite element solutions in various norms. Some computational results are presented to demonstrate the robustness, reliability, accuracy, and flexibility of the WG method for the Brinkman equations.

scholarly journals Local Sobolev estimates of a function by means of its Radon transform

2010 ◽  
Vol 4 (4) ◽  
pp. 721-734 ◽  
Author(s):  
Hans Rullgård ◽  
◽  
Eric Todd Quinto ◽  
Keyword(s):  
Radon Transform ◽  
Sobolev Estimates

scholarly journals Exploiting subtle structural differences in heavy-atom derivatives for experimental phasing

2014 ◽  
Vol 70 (7) ◽  
pp. 1873-1883 ◽  
Author(s):  
Jimin Wang ◽  
Yue Li ◽  
Yorgo Modis
Keyword(s):  
Structure Determination ◽  
Heavy Atom ◽  
Data Sets ◽  
Diarrhea Virus ◽  
Isomorphous Replacement ◽  
Experimental Phasing ◽  
Structural Differences ◽  
Structure Determinations ◽  
Weak Derivatives ◽  
Viral Diarrhea Virus

Structure determination using the single isomorphous replacement (SIR) or single-wavelength anomalous diffraction (SAD) methods with weak derivatives remains very challenging. In a recent structure determination of glycoprotein E2 from bovine viral diarrhea virus, three isomorphous uranium-derivative data sets were merged to obtain partially interpretable initial experimental maps. Small differences between them were then exploited by treating them as three independent SAD data sets plus three circular pairwise SIR data sets to improve the experimental maps. Here, how such subtle structural differences were exploited for experimental phasing is described in detail. The basis for why this approach works is also provided: the effective resolution of isomorphous signals between highly isomorphous derivatives is often much higher than the effective resolution of the anomalous signals of individual derivative data sets. Hence, the new phasing approaches outlined here will be generally applicable to structure determinations involving weak derivatives.

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