10 research outputs found
A Robust Solver for a Second Order Mixed Finite Element Method for the Cahn-Hilliard Equation
We develop a robust solver for a second order mixed finite element splitting
scheme for the Cahn-Hilliard equation. This work is an extension of our
previous work in which we developed a robust solver for a first order mixed
finite element splitting scheme for the Cahn-Hilliard equaion. The key
ingredient of the solver is a preconditioned minimal residual algorithm (with a
multigrid preconditioner) whose performance is independent of the spacial mesh
size and the time step size for a given interfacial width parameter. The
dependence on the interfacial width parameter is also mild.Comment: 17 pages, 3 figures, 4 tables. arXiv admin note: substantial text
overlap with arXiv:1709.0400
Convergence Analysis and Error Estimates for a Second Order Accurate Finite Element Method for the Cahn-Hilliard-Navier-Stokes System
In this paper, we present a novel second order in time mixed finite element
scheme for the Cahn-Hilliard-Navier-Stokes equations with matched densities.
The scheme combines a standard second order Crank-Nicholson method for the
Navier-Stokes equations and a modification to the Crank-Nicholson method for
the Cahn-Hilliard equation. In particular, a second order Adams-Bashforth
extrapolation and a trapezoidal rule are included to help preserve the energy
stability natural to the Cahn-Hilliard equation. We show that our scheme is
unconditionally energy stable with respect to a modification of the continuous
free energy of the PDE system. Specifically, the discrete phase variable is
shown to be bounded in and the discrete
chemical potential bounded in , for any time
and space step sizes, in two and three dimensions, and for any finite final
time . We subsequently prove that these variables along with the fluid
velocity converge with optimal rates in the appropriate energy norms in both
two and three dimensions.Comment: 33 pages. arXiv admin note: text overlap with arXiv:1411.524
A robust solver for a second order mixed finite element method for the CahnāHilliard equation
We develop a robust solver for a second order mixed finite element splitting scheme for the CahnāHilliard equation. This work is an extension of our previous work in which we developed a robust solver for a first order mixed finite element splitting scheme for the CahnāHilliard equation. The key ingredient of the solver is a preconditioned minimal residual algorithm (with a multigrid preconditioner) whose performance is independent of the spatial mesh size and the time step size for a given interfacial width parameter. The dependence on the interfacial width parameter is also mild
Perspectives on ENCODE
The Encylopedia of DNA Elements (ENCODE) Project launched in 2003 with the long-term goal of developing a comprehensive map of functional elements in the human genome. These included genes, biochemical regions associated with gene regulation (for example, transcription factor binding sites, open chromatin, and histone marks) and transcript isoforms. The marks serve as sites for candidate cis-regulatory elements (cCREs) that may serve functional roles in regulating gene expression1. The project has been extended to model organisms, particularly the mouse. In the third phase of ENCODE, nearly a million and more than 300,000 cCRE annotations have been generated for human and mouse, respectively, and these have provided a valuable resource for the scientific community.11Nsciescopu
Expanded encyclopaedias of DNA elements in the human and mouse genomes
AbstractThe human and mouse genomes contain instructions that specify RNAs and proteins and govern the timing, magnitude, and cellular context of their production. To better delineate these elements, phase III of the Encyclopedia of DNA Elements (ENCODE) Project has expanded analysis of the cell and tissue repertoires of RNA transcription, chromatin structure and modification, DNA methylation, chromatin looping, and occupancy by transcription factors and RNA-binding proteins. Here we summarize these efforts, which have produced 5,992 new experimental datasets, including systematic determinations across mouse fetal development. All data are available through the ENCODE data portal (https://www.encodeproject.org), including phase II ENCODE1 and Roadmap Epigenomics2 data. We have developed a registry of 926,535 human and 339,815 mouse candidate cis-regulatory elements, covering 7.9 and 3.4% of their respective genomes, by integrating selected datatypes associated with gene regulation, and constructed a web-based server (SCREEN; http://screen.encodeproject.org) to provide flexible, user-defined access to this resource. Collectively, the ENCODE data and registry provide an expansive resource for the scientific community to build a better understanding of the organization and function of the human and mouse genomes.11Nsciescopu