Discontinuous Galerkin Methods

In this paper, we present an overview of the evolution of the discontinuous Galerkin methods since their introduction in 1973 by Reed and Hill, in the framework of neutron transport, until their most recent developments. We show how these methods made their way into the main stream of computational fluid dynamics and how they are quickly finding use in a wide variety of applications. We review the theoretical and algorithmic aspects of these methods as well as their applications to equations including nonlinear conservation laws, the compressible Navier-Stokes equations, and Hamilton-Jacobi-like equations.

The discontinuous Galerkin method is seemingly immune to many of the problems that commonly plague high-order finite-difference methods, and as such, has the potential to bring the robustness of low-order methods and the efficiency of high-order methods to bear on a broad class of engineering problems. However the dependence of the method on numerical quadrature has significantly increased the cost of the method and limited the use of the method to element shapes for which quadrature formulas are readily available. A quadrature-free formulation has been proposed that allows the discontinuous Galerkin method to be implemented for any element shape and for polynomial basis functions of any degree.

Simplified forms of the space-time discontinuous Galerkin (DG) and discontinuous Galerkin least-squares (DGLS) finite element method are developed and analyzed. The new formulations exploit simplifying properties of entropy endowed conservation law systems while retaining the favorable energy properties associated with symmetric variable formulations.

An implicit high order accurate Discontinuous Galerkin method for the numerical solution of the compressible Favre—Reynolds Averaged Navier—Stokes equations is presented. The method is characterized by a highly compact discretization support even for higher order approximations and this feature can be exploited in the development of implicit integration schemes. Turbulence effects are accounted for by means of the low-Reynolds k-ω turbulence model. A non-standard implementation of the model, whereby the logarithm of ω rather than ω itself is used as unknown, has been found very useful to enhance the stability of the method especially for the higher (third and fourth) order approximations. We present computational results of the transitional flow over a flat plate and of the turbulent flow through a turbine vane with wall heat transfer.

We provide a common framework for the understanding, comparison, and analysis of several discontinuous Galerkin methods that have been proposed for the numerical treatment of elliptic problems. This class includes the recently introduced methods of Bassi and Rebay (together with the variants proposed by Brezzi, Manzini, Marini, Pietra and Russo), the local discontinuous Galerkin methods of Cockburn and Shu, and the method of Baumann and Oden. It also includes the so-called interior penalty methods developed some time ago by Douglas and Dupont, Wheeler, Baker, and Arnold among others.

We summarize several techniques of analysis for finite element methods for linear hyperbolic problems, illustrating their key properties on the simplest model problem. These include the discontinuous Galerkin method, the continuous Galerkin methods on rectangles and triangles, and a nonconforming linear finite element on a special triangular mesh.

We develop software tools for the solution of conservation laws using parallel adaptive discontinuous Galerkin methods. In particular, the Rensselaer Partition Model (RPM) provides parallel mesh structures within an adaptive framework to solve the Euler equations of compressible flow by a discontinuous Galerkin method (LOCO). Results are presented for a Rayleigh-Taylor flow instability for computations performed on 128 processors of an IBM SP computer. In addition to managing the distributed data and maintaining a load balance, RPM provides information about the parallel environment that can be used to tailor partitions to a specific computational environment.

The problem of determining the steady state flow of granular materials in silos under the action of gravity is considered. In the case of a Mohr-Coulomb material, the stress equations correspond to a system of hyperbolic conservation laws with source terms and nonlinear boundary conditions. A higher order Discontinuous Galerkin method is proposed and implemented for the numerical resolution of those equations. The efficiency of the approach is illustrated by the computation of the stress fields induced in silos with sharp changes of the wall angle.

The Discontinuous Galerkin Method (DGM) and Continuous Galerkin Method (CGM) are investigated and compared for the advection problem and the diffusion problem. First, error estimates for Stabilized Discontinuous Galerkin Methods (SDGMs) are presented. Then, conservation laws are discussed for the DGM and CGM. An advantage ascribed to the DGM is the local flux conservation property. It is remarked that the CGM is not only globally conservative, but locally conservative too when a simple post-processing procedure is used. Next, the robustness of different DGMs is investigated numerically. Lastly, the efficiency of the DGM and CGM is compared.

The RKDG method has been effectively used in modeling and simulating semiconductor devices, where the underlying models are hydrodynamic in nature. These include classical as well as quantum models. In this paper, we survey and interpret some of these results. For classical transport, we review the simulation of a benchmark MESFET transistor by means of discontinuous Galerkin methods of degree one. For quantum transport, we report the success in simulation of the resonant tunneling diode. The principal features here are negative differential resistance and hysteresis.

Approximations to solutions of the inhomogeneous boundary value problem for the Navier-Stokes equations are constructed via the discontinuous Galerkin method. The velocity field is approximated using piecewise polynomial functions that are totally discontinuous across interelement boundaries and which are pointwise divergence-free on each element (locally solenoidal). The pressure is approximated by standard continuous piecewise polynomial functions.

The error analysis for finite element methods for partial differential equations can be reduced to estimates of a few integral functionals. Some standard estimates obtained by means of a crude application of the Schwarz inequality do not capture the full order of accuracy of the method. By using a suitable integral identity, however, we can capture the full order of convergence when the mesh is nearly uniform and the exact solution is smooth enough. In this paper, we consider first order linear hyperbolic systems and show how to obtain full order of convergence for the standard Galerkin method and the streamline diffusion method; we also show how to obtain superconvergence in the derivative for the discontinuous Galerkin method. Finally, we show how to obtain full order of convergence for the Ying method for nonlinear scalar conservation laws.

An hp—adaptive conservative Discontinuous Galerkin Method for the solution of convection-diffusion problems is reviewed. A distinctive feature of this method is the treatment of diffusion terms with a new variational formulation. This new variational formulation is not based on mixed formulations, thus having the advantage of not using flux vairables or extended stencils and/or global matrices’ bandwidth when the flux variables are statically condensed at element level.The variational formulation for diffusion terms produces a compact, locally conservative, higher-order accurate, and stable solver. The method supports h—, p—, and hp—approximations and can be applied to any type of domain discretization, including non-matching meshes. A priori error estimates and numerical experiments indicate that the method is robust and capable of delivering high accuracy.

We present an implicit solution method for the compressible Navier-Stokes equations based on a Discontinuous Galerkin space discretization and on the implicit backward Euler time integration scheme. The linear system arising from the implicit time stepping scheme are solved with the preconditioned GMRES iterative method. Several preconditioners have been considered. We describe the features of the method and investigate its accuracy and performance by computing several classical 2-dimensional test cases.

Our focus is on explicit finite element discretization of transient, linear hyperbolic systems in arbitrarily many space dimensions. We propose several ways of generating suitable “explicit” meshes, and sketch an O(hn+1/2) error estimate for a discontinuous Galerkin method. Continuous methods are also considered briefly. This paper parallels [2] in large part, while using a different approach in the analysis.

We develop the error analysis for the hp-version of a discontinuous finite element approximation to second-order partial differential equations with non-negative characteristic form. This class of equations includes classical examples of second-order elliptic and parabolic equations, first-order hyperbolic equations, as well as equations of mixed type. We establish an a priori error bound for the method which is of optimal order in the mesh size h and 1 order less than optimal in the polynomial degree p. In the particular case of a first-order hyperbolic equation the error bound is optimal in h and 1/2 an order less than optimal in p.

Semi-discrete and a family of discrete time locally conservative Discontinuous Galerkin procedures are formulated for approximations to nonlinear parabolic equations. For the continuous time approximations a priori L∞(L2) and L2(Hl) estimates are derived and similarly, l∞ (L2) and l2 (H1) for the discrete time schemes. Spatial rates in Hl and time truncation errors in L2 are optimal.

We compare an iterative asynchronous parallel algorithm for the solution of partial differential equations, with a synchronous algorithm, in terms of termination detection schemes and performance. Both algorithms are based on discontinuous Galerkin finite-element methods, in which the local elements provide a natural decomposition of the problem into computationally-independent sets. We demonstrate the superiority of the asynchronous algorithm over the synchronous one in terms of the overall execution time. Our goal is to persuade parallel developers that it is worthwhile to implement the more complex asynchronous algorithm.

We present a class of numerical schemes for the numerical integration of first order Hamilton Jacobi equations. The method can be considered as Discontinuous Galerkin scheme, the viscosity solution is directly adapted into the numerical scheme, contrary to other authors.

To date, the more successful numerical methods in viscoelastic fluid dynamics are based upon the so called Discrete Elastic Viscous Stress Splitting (DEVSS) algorithm [6] together with a suitable form of upwinding of the hyperbolic part of the constitutive equation. An elegant way to perform upwinding on the viscoelastic stress tensor can be found in Discontinuous Galerkin techniques [4]. In particular the recently developed DEVSS/DG version [1], has proven to be successful in analyzing viscoelastic fluid flow problems in both smooth and non-smooth geometries. A particularly attractive feature of DG-based methods is that they allow for an efficient resolution of flow problems with multiple relaxation times, as was demonstrated in Baaijens et al. [1] which has recently been extended to three dimensional flows [2].However, one of the key issues in simulations of viscoelastic flows remains the assessment of temporal stability of the computational method. Especially, increasing elasticity beyond critical values of the Weissenberg number can give rise to numerical instabilities in flows that are otherwise mathematically stable.

We construct high order current vector basis functions on an arbitrary curved surface. The objective is to construct vector basis functions which consist of high order polynomials of the surface parameterization variables on curved triangles and have continuous normal components. Explicit formulation of high order current basis functions is provided.

This paper presents an adaptive finite element model for oxidation-driven fracture that uses space-time elements to track continuous crack-tip motion. The model incorporates viscoplastic material behavior, stress-enhanced diffusive transport of reactive chemical species and a cohesive interface fracture criterion. We discuss the weak formulation of the coupled system, including stabilized discontinuous Galerkin formulations for the chemical diffusion and the material evolution equations.

L2 error estimates for the Local Discontinuous Galerkin (LDG) method have been theoretically proven for linear convection diffusion problems and periodic boundary conditions. It has been proven that when polynomials of degree k are used, the LDG method has a suboptimal order of convergence k. However, numerical experiments show that under a suitable choice of the numerical flux, higher order of convergence can be achieved. In this paper, we consider Dirichlet boundary conditions and we show that the LDG method has an optimal order of convergence k + 1.

It is well known that the discontinuous Galerkin (DG) method for scalar linear conservation laws has an order of convergence of k + 1/2 when polynomials of degree k are used and the exact solution is sufficiently smooth. In this paper, we show that a suitable post-processing of the DG approximate solution is of order 2k+1 in L2(Ω₀) where Ω₀ is a domain on which the exact solution is smooth enough. The post-processing is a convolution with a kernel whose support has measure of order one if the meshes are arbitrary; if the meshes are translation invariant, the support of the kernel is a cube whose edges are of size of order ∆x only. The post-processing has to be performed only once, at the final time level.

Wavelets provide a tool for efficient representation of functions. This efficient representation has proven useful in the numerical solution of non-linear evolution equations. In this paper we provide a brief review of the use of wavelets for efficient representation of functions, and in particular we describe the piecewise-discontinuous basis of wavelets proposed by Alpert. We review the useful properties this basis has for the solution of PDE’s, and introduce an illustrative approach to the representation of boundary conditions. We also discuss the extension to higher dimensional problems.

We discuss the application of the Local Discontinuous Galerkin method to the approximation of contaminant transport in porous media.

This paper is devoted to the presentation of a new family of high order numerical schemes in space (we restrict the scope of this paper to first order in time) for the numerical solution of Euler equations in axisymmetric geometry (1) (see also [Des98a] in 1D and [Des98b]). The unknowns are the density, the two components of the velocity and the specific total energy (ρ, u₁, u₂, e).

The numerical simulation of polymer processing problems requires important computer ressources making essential the development of very efficient numerical techniques. Among the difficulties, the viscoelastic nature of polymers is the most important. Nonlinear multi-mode differential models exist to describe their behaviour, but their hyperbolic (convective) nature makes numerical simulations quite difficult. The presence of free surfaces in problems such as injection molding process, extrusion die swell, coextrusion, etc. further enhance the complexity.

We consider discontinuous in time and continuous in space Galerkin finite-element methods for the numerical solution of reaction-diffusion differential equations. These are implicit methods that require the solution of a system of nonlinear equations at each time node. In this paper, we explore the use of Krylov-subspace techniques for the iterative solution of the linear systems that arise when these nonlinear systems are solved by means of Newton-type methods. It is shown how these linear systems depend on the choice of the basis functions used for the time discretization. We demonstrate that Krylov-subspace methods can be sped up considerably by employing an orthogonal basis for the time discretization and by combining the Krylov iteration with a suitable block preconditioner. Results of numerical experiments are reported.

This brief account will call attention to a line of research which stretches over 40 years, and which appears now to be joining the mainstream of work on the discretization of linear partial differential equations. The first in a series of papers [1] describing the Method of Cells, now called the Cell Discretization Algorithm (CD or CDA), was published in 1959.

In this presentation, we perform further investigation on the least square procedure used in the discontinuous Galerkin methods developed in [2] and [3] for the two-dimensional Hamilton-Jacobi equations. The focus of this presentation will be upon the influence of this least square procedure to the accuracy and stability of the numerical results. We will show through numerical examples that the procedure is crucial for the success of the discontinuous Galerkin methods developed in [2] and [3], especially for high order methods. New test cases using P4 polynomials, which are at least fourth order and often fifth order accurate, are shown, in addition to the P2 and P3 cases presented in [2] and [3]. This addition is non-trivial as the least square procedure plays a more significant role for the P4 case.

We derive a posteriori error estimates for the nonconforming rotated bilinear element. The estimates are residual based and make use of weight factors obtained by a duality argument. Galerkin orthogonality requires the introduction of additional local trial functions. We show that their influence is of higher order and that they can be neglected. The validity of the estimate is demonstrated by computations for the Laplacian and for Stokes’ equations.

Two discontinuous spectral element methods for the solution of Maxwell’s equations are compared. The first method is a staggered-grid Chebyshev approximation. The second is a spectral element (collocation) form of the discontinuous Galerkin method. In both methods, the approximations are discontinuous at element boundaries, making them suitable for propagating waves through multiple materials. Solutions are presented for propagation of a plane wave through a plane dielectric interface, and for scattering off a coated perfectly conducting cylinder.

This article considers a posteriori error estimation of specified functionals for first-order systems of conservation laws discretized using the discontinuous Galerkin (DG) finite element method. Using duality techniques, we derive exact error representation formulas for both linear and nonlinear functionals given an associated bilinear or nonlinear variational form. Weighted residual approximations of the exact error representation formula are then proposed and numerically evaluated for Ringleb flow, an exact solution of the 2-D Euler equations.

In this presentation we explore a recently introduced high order discontinuous Galerkin method for two dimensional incompressible flow in vorticity streamfunction formulation [8]. In this method, the momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin method. The streamfunction is obtained by a standard Poisson solver using continuous finite elements. There is a natural matching between these two finite element spaces, since the normal component of the velocity field is continuous across element boundaries. This allows for a correct upwinding gluing in the discontinuous Galerkin framework, while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability. The method is suitable for inviscid or high Reynolds number flows. In our previous work, optimal error estimates are proven and verified by numerical experiments. In this presentation we present one numerical example, the shear layer problem, in detail and from different angles to illustrate the resolution performance of the method.

We present a matrix-free discontinuous Galerkin method for simulating compressible viscous flows in two- and three-dimensional moving domains in an Arbitrary Lagrangian Eulerian (ALE) framework. Spatial discretization is based on standard structured and unstructured grids but using an orthogonal spectral hierarchical basis. The method is third-order accurate in time, and converges exponentially fast in space for smooth solutions. We also report on open issues related to quadrature crimes and over-integration.

A Discontinuous Galerkin method is applied to hyperbolic systems that contain stiff relaxation terms. We demonstrate that when the relaxation time is unresolved, the method is accurate in the sense that it accurately represents the system’s Chapman-Enskog approximation. Results are presented for the hyperbolic heat equation and coupled radiation-hydrodynamics.

We present a Neumann-subproblem a posteriori finite element procedure for the efficient calculation of rigorous, constant-free, sharp lower and upper estimators for linear functional outputs of parabolic equations discretized by a discontinuous Galerkin method in time. We first formulate the bound procedure; we then provide illustrative numerical examples for problems of unsteady heat conduction.

We describe a numerical scheme to solve 3D Arbitrary Lagrangian-Eulerian (ALE) hydrodynamics on an unstructured mesh using a discontinuous Galerkin method (DGM) and an explicit Runge-Kutta time discretization. Upwinding is achieved through Roe’s linearized Riemann solver with the Harten-Hyman entropy fix. For stabilization, a 3D quadratic programming generalization of van Leer’s 1D minmod slope limiter is used along with a Lapidus type artificial viscosity. This DGM scheme has been tested on a variety of hydrodynamic test problems and appears to be robust making it the basis for the integrated 3D inertial confinement fusion modeling code ICF3D. For efficient code development, we use C₊₊ object oriented programming to easily separate the complexities of an unstructured mesh from the basic physics modules. ICF3D is fully parallelized using domain decomposition and the MPI message passing library. It is fully portable. It runs on uniprocessor workstations and massively parallel platforms with distributed and shared memory.

The discontinuous Galerkin methods have recently found increasing applications in computational fluid dynamics because of their robustness and other practical features. The key feature that distinguishes the discontinuous spectral Galerkin method from its traditional counterpart is that the basis functions in each element are independent of the basis functions in the contiguous elements. The method is thus compact, and it easily accommodates any boundary conditions and complex geometry.

In [S1], a method for the numerical approximation of singularly perturbed convection diffusion problems was introduced. In this note, we will show an a posteriori error estimate for this method.

The authors present a numerical solution for the shallow water equations based on the Runge Kutta Discontinuous Galerkin method. Modeling sink and source terms introduces restrictions to the space discretization and a modification of the slope limiter. Hydraulic test problems and a real-world application show the good performance of the scheme.

The dispersion relation of the semi-discrete continuous and discontinuous Galerkin formulations are analysed for the linear advection equation. In the context of an spectral/hp element discretisation on an equispaced mesh the problem can be reduced to a P × P eigenvalue problem where P is the polynomial order. The analytical dispersion relationships for polynomial order up to P = 3 and the numerical values for P = 10 are presented demonstrating similar dispersion properties but show that the discontinuous scheme is more diffusive.

This non-conforming extension of the finite element method is illustrated with a model elliptic problem and other applications are sketched. New results concerning domain decomposition and the construction of a solenoidal basis for the Stokes equations are described.

An analysis of the balance between the computational complexity, accuracy, and resolution requirements of a discontinuous Galerkin finite element method for the solution of the compressible Euler equations of gas dynamics is presented. The discontinuous Galerkin finite element method uses a very local discretization, which remains second order accurate on highly non-uniform meshes, but at the cost of an increase in computational complexity and memory use. The question of the balance between computational complexity and accuracy is addressed by studying the evolution of vortices in the wake of a wing. It is demonstrated that the discontinuous Galerkin finite element method on locally refined meshes can result in a significant reduction in computational cost.

An Eulerian-Lagrangian localized adjoint method (ELLAM) is presented for coupled systems of fluid flow processes occurring in porous media with point sources and sinks. The ELLAM scheme symmetrizes the governing transport equation, greatly eliminates non-physical oscillation and/or excessive numerical dispersion present in many large-scale simulators widely used in industrial applications. It can treat large mobility ratios, discontinuous permeabilities and porosities, anisotropic dispersion in tensor form, and point sources and sinks. It also conserves mass. Numerical experiments are presented. The relationship between ELLAM and discontinuous Galerkin methods (DGMs), and the possibility of developing a hybrid ELLAM-DGM scheme are discussed.

In this paper we demonstrate the efficiency of using the discontinuous Galerkin method for simulating electromagnetic scattering problems using Maxwell’s equations. We show that it is possible to use unstructured hp-finite elements in mixed-element (polymorphic) grids. We include examples of scattering from a two-dimensional cylinder and preliminary results from a three-dimensional F15 geometry.

We present a new space-time discontinuous Galerkin formulation for elastodynamics. The method allows for jumps in the field variables across inter-element boundaries with arbitrary orientation. The resulting method is locally conservative and admits a direct element-by-element solution procedure.

Both nonconforming and enhanced strain methods are analyzed under the framework of the mixed method. The notion of selective nonconforming or selective enhanced strain methods are introduced.