Graham F. Carey

Bio: Graham F. Carey is an academic researcher from University of Texas at Austin. The author has contributed to research in topics: Finite element method & Mixed finite element method. The author has an hindex of 37, co-authored 253 publications receiving 5803 citations. Previous affiliations of Graham F. Carey include University of Texas System.

libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations

Abstract: In this paper we describe the libMesh (http://libmesh.sourceforge.net) framework for parallel adaptive finite element applications. libMesh is an open-source software library that has been developed to facilitate serial and parallel simulation of multiscale, multiphysics applications using adaptive mesh refinement and coarsening strategies. The main software development is being carried out in the CFDLab (http://cfdlab.ae.utexas.edu) at the University of Texas, but as with other open-source software projects; contributions are being made elsewhere in the US and abroad. The main goals of this article are: (1) to provide a basic reference source that describes libMesh and the underlying philosophy and software design approach; (2) to give sufficient detail and references on the adaptive mesh refinement and coarsening (AMR/C) scheme for applications analysts and developers; and (3) to describe the parallel implementation and data structures with supporting discussion of domain decomposition, message passing, and details related to dynamic repartitioning for parallel AMR/C. Other aspects related to C++ programming paradigms, reusability for diverse applications, adaptive modeling, physics-independent error indicators, and similar concepts are briefly discussed. Finally, results from some applications using the library are presented and areas of future research are discussed.

High‐order compact scheme for the steady stream‐function vorticity equations

Abstract: A higher-order compact scheme that is O(h4) on the nine-point 2-D stencil is formulated for the steady stream-function vorticity form of the Navier-Stokes equations. The resulting stencil expressions are presented and hence this new scheme can be easily incorporated into existing industrial software. We also show that special treatment of the wall boundary conditions is required. The method is tested on representative model problems and compares very favourably with other schemes in the literature.

Least-squares mixed finite elements for second-order elliptic problems

Abstract: A theoretical analysis of a least-squares mixed finite element method for second-order elliptic problems in two- and three-dimensional domains is presented. It is proved that the method is not subj...

Resonant Phase Patterns in a Reaction-Diffusion System

Abstract: Resonance regions similar to the Arnol'd tongues found in single oscillator frequency locking are observed in experiments using a spatially extended periodically forced Belousov-Zhabotinsky system. We identify six distinct 2:1 subharmonic resonant patterns and describe them in terms of the position-dependent phase and magnitude of the oscillations. Some experimentally observed features are also found in numerical studies of a forced Brusselator reaction-diffusion model.

Machine learning

Abstract: Machine Learning is the study of methods for programming computers to learn. Computers are applied to a wide range of tasks, and for most of these it is relatively easy for programmers to design and implement the necessary software. However, there are many tasks for which this is difficult or impossible. These can be divided into four general categories. First, there are problems for which there exist no human experts. For example, in modern automated manufacturing facilities, there is a need to predict machine failures before they occur by analyzing sensor readings. Because the machines are new, there are no human experts who can be interviewed by a programmer to provide the knowledge necessary to build a computer system. A machine learning system can study recorded data and subsequent machine failures and learn prediction rules. Second, there are problems where human experts exist, but where they are unable to explain their expertise. This is the case in many perceptual tasks, such as speech recognition, hand-writing recognition, and natural language understanding. Virtually all humans exhibit expert-level abilities on these tasks, but none of them can describe the detailed steps that they follow as they perform them. Fortunately, humans can provide machines with examples of the inputs and correct outputs for these tasks, so machine learning algorithms can learn to map the inputs to the outputs. Third, there are problems where phenomena are changing rapidly. In finance, for example, people would like to predict the future behavior of the stock market, of consumer purchases, or of exchange rates. These behaviors change frequently, so that even if a programmer could construct a good predictive computer program, it would need to be rewritten frequently. A learning program can relieve the programmer of this burden by constantly modifying and tuning a set of learned prediction rules. Fourth, there are applications that need to be customized for each computer user separately. Consider, for example, a program to filter unwanted electronic mail messages. Different users will need different filters. It is unreasonable to expect each user to program his or her own rules, and it is infeasible to provide every user with a software engineer to keep the rules up-to-date. A machine learning system can learn which mail messages the user rejects and maintain the filtering rules automatically. Machine learning addresses many of the same research questions as the fields of statistics, data mining, and psychology, but with differences of emphasis. Statistics focuses on understanding the phenomena that have generated the data, often with the goal of testing different hypotheses about those phenomena. Data mining seeks to find patterns in the data that are understandable by people. Psychological studies of human learning aspire to understand the mechanisms underlying the various learning behaviors exhibited by people (concept learning, skill acquisition, strategy change, etc.).

The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique

Abstract: This is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation. In this paper, a general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes. The implementation of the recovery technique is simple and cost effective. The technique has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems. Numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent. One order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements. In particular, an O(h4) convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time. The performance of the proposed technique is compared with the widely used smoothing procedure of global L2 projection and other methods. It is found that the derivatives recovered at interelement nodes, by using L2 projection, are also superconvergent for linear elements but not for quadratic elements. Numerical experiments on the convergence of the recovered solutions in the energy norm are also presented. Higher rates of convergence are again observed. The results presented in this part of the paper indicate clearly that a new, powerful and economical process is now available which should supersede the currently used post-processing procedures applied in most codes.

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources

Abstract: The authors develop finite difference methods for elliptic equations of the form \[ abla \cdot (\beta (x) abla u(x)) + \kappa (x)u(x) = f(x)\] in a region $\Omega $ in one or two space dimension...