This volume contains current progress of a new class of finite element method, the Discontinuous Galerkin Method (DGM), which has been under rapid developments recently and has found its use very quickly in such diverse applications as aeroacoustics, semi-conductor device simulation, turbomachinery, turbulent flows, materials processing, Magneto-hydro-dynamics, 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 effect in organizing and publishing the existing volume of knowledge on this subject. The current volume organizes this knowledge and it covers both theoretical as well as practical issues of the Discontinuous Galerkin method.

I Overview.- The Development of Discontinuous Galerkin Methods.- II Invited Papers.- Steps Toward a Robust High-Order Simulation Tool for Aerospace Applications.- Simplified Discontinuous Galerkin Methods for Systems of Conservation Laws with Convex Extension.- A High Order Discontinuous Galerkin Method for Compressible Turbulent Flows.- Discontinuous Galerkin Methods for Elliptic Problems.- Analysis of Finite Element Methods for Linear Hyperbolic Problems.- Software for the Parallel Adaptive Solution of Conservation Laws by Discontinuous Galerkin Methods.- Simulation of Gravity Flow of Granular Materials in Silos.- A Comparison of Discontinuous and Continuous Galerkin Methods Based on Error Estimates, Conservation, Robustness and Efficiency.- The Utility of Modeling and Simulation in Determining Transport Performance Properties of Semiconductors.- A Discontinuous Galerkin Method for the Incompressible Navier-Stokes Equations.- Full Convergence for Hyperbolic Finite Elements.- A Conservative DGM for Convection-Diffusion and Navier-Stokes Problems.- GMRES Discontinuous Galerkin Solution of the Compressible Navier-Stokes Equations.- Explicit Finite Element Methods for Linear Hyperbolic Systems.- hp-DGFEM for Partial Differential Equations with Nonnegative Characteristic Form.- A Discontinuous Galerkin Method Applied to Nonlinear Parabolic Equations.- III Contributed Papers.- Parallel Iterative Discontinuous Galerkin Finite-Element Methods.- A Discontinuous Projection Algorithm for Hamilton Jacobi Equations.- Successes and Failures of Discontinuous Galerkin Methods in Viscoelastic Fluid Analysis.- High Order Current Basis Functions for Electromagnetic Scattering of Curved Surfaces.- An Adaptive Discontinuous Galerkin Model for Coupled Viscoplastic Crack Growth and ChemicalTransport.- An Optimal Estimate for the Local Discontinuous Galerkin Method.- Post-Processing of Galerkin Methods for Hyperbolic Problems.- to Discontinuous Wavelets.- The Local Discontinuous Galerkin Method for Contaminant Transport Problems.- Discontinuous Galerkin Method for the Numerical Solution of Euler Equations in Axisymmetric Geometry.- Ten Years Using Discontinuous Galerkin Methods for Polymer Processing Problems.- Using Krylov-Subspace Iterations in Discontinuous Galerkin Methods for Nonlinear Reaction-Diffusion Systems.- An Abridged History of Cell Discretization.- The Effect of the Least Square Procedure for Discontinuous Galerkin Methods for Hamilton-Jacobi Equations.- A Posteriori Error Estimate in the Case of Insufficient Regularity of the Discrete Space.- Discontinuous Spectral Element Approximation of Maxwell's Equations.- A Posteriori Error Estimation for Adaptive Discontinuous Galerkin Approximations of Hyperbolic Systems.- A Numerical Example on the Performance of High Order Discontinuous Galerkin Method for 2D Incompressible Flows.- A Discontinuous Galerkin Method in Moving Domains.- Discontinuous Galerkin for Hyperbolic Systems with Stiff Relaxation.- Finite Element Output Bounds for Parabolic Equations: Application to Heat Conduction Problems.- 3D Unstructured Mesh ALE Hydrodynamics with the Upwind Discontinuous Galerkin Method.- Some Remarks on the Accuracy of a Discontinuous Galerkin Method.- Coupling Continuous and Discontinuous Techniques: An Adaptive Approach.- A Discontinuous Galerkin Method for the Shallow Water Equations with Source Terms.- Dispersion Analysis of the Continuous and Discontinuous Galerkin Formulations.- The Cell Discretization Algorithm; An Overview.- Accuracy, Resolution, and Computational Complexity of a Discontinuous Galerkin Finite Element Method.- An ELLAM Scheme for Porous Medium Flows.- Application of the Discontinuous Galerkin Method to Maxwell's Equations Using Unstructured Polymorphic hp-Finite Elements.- A Space-Time Discontinuous Galerkin Method for Elastodynamic Analysis.- Nonconforming, Enhanced Strain, and Mixed Finite Element Methods - A Unified Approach.