Discontinuous Galerkin Methods

1. Front Matter

   Pages I-XI

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2. Overview

   1. Front Matter

      Pages 1-1

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   2. The Development of Discontinuous Galerkin Methods

      • Bernardo Cockburn, George E. Karniadakis, Chi-Wang Shu

      Pages 3-50

3. Invited Papers

   1. Front Matter

      Pages 51-51

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   2. Steps Toward a Robust High-Order Simulation Tool for Aerospace Applications

      • Harold L. Atkins

      Pages 53-61

   3. Simplified Discontinuous Galerkin Methods for Systems of Conservation Laws with Convex Extension

      • Timothy J. Barth

      Pages 63-75

   4. A High Order Discontinuous Galerkin Method for Compressible Turbulent Flows

      • F. Bassi, S. Rebay

      Pages 77-88

   5. Discontinuous Galerkin Methods for Elliptic Problems

      • Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, Donatella Marini

      Pages 89-101

   6. Analysis of Finite Element Methods for Linear Hyperbolic Problems

      • Richard S. Falk

      Pages 103-112

   7. Software for the Parallel Adaptive Solution of Conservation Laws by Discontinuous Galerkin Methods

      • J. E. Flaherty, R. M. Loy, M. S. Shephard, J. D. Teresco

      Pages 113-123

   8. Simulation of Gravity Flow of Granular Materials in Silos

      • Pierre A. Gremaud, John V. Matthews

      Pages 125-134

   9. A Comparison of Discontinuous and Continuous Galerkin Methods Based on Error Estimates, Conservation, Robustness and Efficiency

      • Thomas J. R. Hughes, Gerald Engel, Luca Mazzei, Mats G. Larson

      Pages 135-146

   10. The Utility of Modeling and Simulation in Determining Transport Performance Properties of Semiconductors

       • Bernardo Cockburn, Joseph W. Jerome, Chi-Wang Shu

       Pages 147-156

   11. A Discontinuous Galerkin Method for the Incompressible Navier-Stokes Equations

       • Ohannes Karakashian, Theodoros Katsaounis

       Pages 157-166

   12. Full Convergence for Hyperbolic Finite Elements

       • Qun Lin

       Pages 167-177

   13. A Conservative DGM for Convection-Diffusion and Navier-Stokes Problems

       • J. Tinsley Oden, Carlos Erik Baumann

       Pages 179-196

   14. GMRES Discontinuous Galerkin Solution of the Compressible Navier-Stokes Equations

       • F. Bassi, S. Rebay

       Pages 197-208

   15. Explicit Finite Element Methods for Linear Hyperbolic Systems

       • Richard S. Falk, Gerard R. Richter

       Pages 209-219

   16. hp-DGFEM for Partial Differential Equations with Nonnegative Characteristic Form

       • Endre Süli, Christoph Schwab, Paul Houston

       Pages 221-230

   17. A Discontinuous Galerkin Method Applied to Nonlinear Parabolic Equations

       • Béatrice Rivière, Mary F. Wheeler

       Pages 231-244

A class of finite element methods, the Discontinuous Galerkin Methods (DGM), has been under rapid development recently and has found its use very quickly in such diverse applications as aeroacoustics, semi-conductor device simula­ tion, turbomachinery, turbulent flows, materials processing, MHD and plasma simulations, and image processing. While there has been a lot of interest from mathematicians, physicists and engineers in DGM, only scattered information is available and there has been no prior effort in organizing and publishing the existing volume of knowledge on this subject. In May 24-26, 1999 we organized in Newport (Rhode Island, USA), the first international symposium on DGM with equal emphasis on the theory, numerical implementation, and applications. Eighteen invited speakers, lead­ ers in the field, and thirty-two contributors presented various aspects and addressed open issues on DGM. In this volume we include forty-nine papers presented in the Symposium as well as a survey paper written by the organiz­ ers. All papers were peer-reviewed. A summary of these papers is included in the survey paper, which also provides a historical perspective of the evolution of DGM and its relation to other numerical methods. We hope this volume will become a major reference in this topic. It is intended for students and researchers who work in theory and application of numerical solution of convection dominated partial differential equations. The papers were written with the assumption that the reader has some knowledge of classical finite elements and finite volume methods.

• Approximation
• conservation laws
• discontinuous approximation
• finite elements
• finite volume
• image processing
• knowledge
• simulation

• School of Mathematics, University of Minnesota, Minneapolis, USA

  Bernardo Cockburn

• Division of Applied Mathematics, Brown University, Providence, USA

  George E. Karniadakis, Chi-Wang Shu

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