We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems.

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Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems

In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobilike equations.

Review Article Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems

In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobi-like equations.

Runge-Kutta discontinuous Galerkin methods for convection-dominated problems

We introduce a unifying framework for hybridization of finite element methods for second order elliptic problems. The methods fitting in the framework are a general class of mixed-dual finite element methods including hybridized mixed, continuous Galerkin, nonconforming, and a new, wide class of hybridizable discontinuous Galerkin methods. The distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements. Since the associated matrix is sparse, symmetric, and positive definite, these methods can be efficiently implemented. Moreover, the framework allows, in a single implementation, the use of different methods in different elements or subdomains of the computational domain, which are then automatically coupled. Finally, the framework brings about a new point of view, thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques, as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom.

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Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems

Finite element methods for Maxwell's equations.

The symmetric interior penalty discontinuous Galerkin finite element method is presented for the numerical discretization of the second‐order wave equation. The resulting stiffness matrix is symmetric positive definite, and the mass matrix is essentially diagonal; hence, the method is inherently parallel and leads to fully explicit time integration when coupled with an explicit time‐ stepping scheme. Optimal a priori error bounds are derived in the energy norm and the $L^2$‐norm for the semidiscrete formulation. In particular, the error in the energy norm is shown to converge with the optimal order ${\cal O}(h^{\min\{s,\ell\}})$ with respect to the mesh size h, the polynomial degree $\ell$, and the regularity exponent s of the continuous solution. Under additional regularity assumptions, the $L^2$‐error is shown to converge with the optimal order ${\cal O}(h^{\min\{s,\ell\}})$. Numerical results confirm the expected convergence rates and illustrate the versatility of the method.

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Discontinuous Galerkin Finite Element Method for the Wave Equation

In this paper, we present a superconvergence result for the local discontinuous Galerkin (LDG) method for a model elliptic problem on Cartesian grids. We identify a special numerical flux for which the L2-norm of the gradient and the L2-norm of the potential are of orders k+1/2 and k+1, respectively, when tensor product polynomials of degree at most k are used; for arbitrary meshes, this special LDG method gives only the orders of convergence of k and k+1/2, respectively. We present a series of numerical examples which establish the sharpness of our theoretical results.

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Superconvergence of the Local Discontinuous Galerkin Method for Elliptic Problems on Cartesian Grids

In this paper a new local discontinuous Galerkin method for the incompressible stationary Navier-Stokes equations is proposed and analyzed. Four important features render this method unique: its stability, its local conservativity, its high-order accuracy, and the exact satisfaction of the incompressibility constraint. Although the method uses completely discontinuous approximations, a globally divergence-free approximate velocity in H(div; Ω) is obtained by simple, element-by-element post-processing. Optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge. This procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible Navier-Stokes equations by solving a sequence of Oseen problems. Numerical results are shown which verify the theoretical rates of convergence. They also confirm the independence of the number of fixed point iterations with respect to the discretization parameters. Finally, they show that the method works well for a wide range of Reynolds numbers.

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A locally conservative LDG method for the incompressible Navier-Stokes equations

We present a class of discontinuous Galerkin methods for the incompressible Navier---Stokes equations yielding exactly divergence-free solutions. Exact incompressibility is achieved by using divergence-conforming velocity spaces for the approximation of the velocities. The resulting methods are locally conservative, energy-stable, and optimally convergent. We present a set of numerical tests that confirm these properties. The results of this note naturally expand the work in Cockburn et al. (2005) Math. Comp. 74, 1067---1095.

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A Note on Discontinuous Galerkin Divergence-free Solutions of the Navier-Stokes Equations

In this paper, we introduce and analyze local discontinuous Galerkin methods for the Stokes system. For a class of shape regular meshes with hanging nodes we derive a priori estimates for the L2-norm of the errors in the velocities and the pressure. We show that optimal-order estimates are obtained when polynomials of degree k are used for each component of the velocity and polynomials of degree k-1 for the pressure, for any $k\ge1$. We also consider the case in which all the unknowns are approximated with polynomials of degree k and show that, although the orders of convergence remain the same, the method is more efficient. Numerical experiments verifying these facts are displayed.

/pdf/local-discontinuous-galerkin-methods-for-the-stokes-system-420twiv4rn.pdf

Dominik Schötzau

Papers

Discontinuous Galerkin Finite Element Method for the Wave Equation

Superconvergence of the Local Discontinuous Galerkin Method for Elliptic Problems on Cartesian Grids

A locally conservative LDG method for the incompressible Navier-Stokes equations

A Note on Discontinuous Galerkin Divergence-free Solutions of the Navier-Stokes Equations

Local Discontinuous Galerkin Methods for the Stokes System