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Joseph E. Flaherty

Bio: Joseph E. Flaherty is an academic researcher from Rensselaer Polytechnic Institute. The author has contributed to research in topics: Finite element method & Mesh generation. The author has an hindex of 41, co-authored 119 publications receiving 4864 citations. Previous affiliations of Joseph E. Flaherty include United States Department of the Army & University of Utah.


Papers
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Journal ArticleDOI
TL;DR: By detecting discontinuities in such variables as density or entropy, limiting may be applied only in these regions; thereby, preserving a high order of accuracy in regions where solutions are smooth.

404 citations

Journal ArticleDOI
TL;DR: This work constructs parallel finite element methods for the solution of hyperbolic conservation laws in one and two dimensions and presents results using adaptive h- and p-refinement to reduce the computational cost of the method.

370 citations

Journal ArticleDOI
TL;DR: Convergence of local and global discretization errors to the Radau polynomial of degree p +1 holds for smooth solutions as p →∞ and is used to construct asymptotically correct a posteriori estimates of spatial discretized errors that are effective for linear and nonlinear conservation laws in regions where solutions are smooth.

226 citations

Journal ArticleDOI
TL;DR: To accommodate the variable time steps, octree partitioning is extended to use weights derived from element size and processor load imbalances are corrected by using traversals of an octree representing a spatial decomposition of the domain.

210 citations

Journal ArticleDOI
TL;DR: A high-order formulation for solving hyperbolic conservation laws using the discontinuous Galerkin method (DGM) is presented and an orthogonal basis for the spatial discretization is introduced and use explicit Runge--Kutta time discretized.
Abstract: We present a high-order formulation for solving hyperbolic conservation laws using the discontinuous Galerkin method (DGM). We introduce an orthogonal basis for the spatial discretization and use explicit Runge--Kutta time discretization. Some results of higher order adaptive refinement calculations are presented for inviscid Rayleigh--Taylor flow instability and shock reflection problems. The adaptive procedure uses an error indicator that concentrates the computational effort near discontinuities.

202 citations


Cited by
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Book
01 Apr 2003
TL;DR: This chapter discusses methods related to the normal equations of linear algebra, and some of the techniques used in this chapter were derived from previous chapters of this book.
Abstract: Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

13,484 citations

Journal ArticleDOI
TL;DR: Gmsh as mentioned in this paper is an open-source 3D finite element grid generator with a build-in CAD engine and post-processor that provides a fast, light and user-friendly meshing tool with parametric input and advanced visualization capabilities.
Abstract: Gmsh is an open-source 3-D finite element grid generator with a build-in CAD engine and post-processor. Its design goal is to provide a fast, light and user-friendly meshing tool with parametric input and advanced visualization capabilities. This paper presents the overall philosophy, the main design choices and some of the original algorithms implemented in Gmsh. Copyright (C) 2009 John Wiley & Sons, Ltd.

5,322 citations

Journal ArticleDOI
TL;DR: To the best of our knowledge, there is only one application of mathematical modelling to face recognition as mentioned in this paper, and it is a face recognition problem that scarcely clamoured for attention before the computer age but, having surfaced, has attracted the attention of some fine minds.
Abstract: to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

3,015 citations

Journal ArticleDOI
TL;DR: It is proven that for scalar equations, the LDG methods are L2-stable in the nonlinear case and in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations.
Abstract: In this paper, we study the local discontinuous Galerkin (LDG) methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge--Kutta discontinuous Galerkin (RKDG) methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, high-order formal accuracy, and easy handling of complicated geometries for convection-dominated problems. It is proven that for scalar equations, the LDG methods are L2-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.

2,265 citations

Book
28 May 1996
TL;DR: Introduction.
Abstract: Introduction. A Simple Model Problem. Abstract Nonlinear Equations. Finite Element Discretizations of Elliptic PDEs. Practical Implementation. Bibliography. Subject Index.

2,253 citations