Lecture# 1: Discontinuous Galerkin Methods: Motivation and Their Origin (original) (raw)
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The presentation is concerned with the numerical treatment of transient transport problems like heat transfer or mass/species transport by means of discontinuous spatial discretization and different time integration schemes. To achieve a semidiscrete initial value problem discontinuous p-finite elements [1] for the approximation in space are used, where the continuity at the interelement boundaries is just weakly enforced. Furthermore, continuous and discontinuous Galerkin time integration schemes, which evaluate the balance equation in a weak sense over the time interval, are presented [7]. These discretization techniques are investigated with respect to robustness, reliability and accuracy, also in the context of non-smooth initial or boundary conditions. Selected benchmark analyses of heat conduction with an available analytical solution are analyzed for the above described numerical methods. Moreover the highly non-linear reaction-diffusion process of calcium leaching in cementi...
SIAM Journal on Numerical Analysis, 2008
We study the approximation of non-smooth solutions of the transport equation in one-space dimension by approximations given by a Runge-Kutta discontinuous Galerkin method of order two. We take an initial data which has compact support and is smooth except at a discontinuity, and show that, if the ratio of the time step size to the grid size is less than 1/3, the error at the time T in the L 2 (R\R T )−norm is the optimal order two when R T is a region of size O(T 1/2 h 1/2 log 1/h) to the right of the discontinuity and of size O(T 1/3 h 2/3 log 1/h) to the left. Numerical experiments validating these results are presented.
Discontinuous Galerkin Methods for Partial Differential Equations
The purpose of this meeting was to bring together researchers in a wide variety of areas working on discontinuous Galerkin (DG) methods for partial differential equations, to investigate and identify problems of current interest and to exchange ideas and viewpoints on the most recent developments of these methods. There were 33 participants, mostly from American and Canadian universities, including students and postdoctoral fellows. The program of the workshop consisted of 28 half-hour talks.
Discontinuous Galerkin methods
Journal of Applied Mathematics and Mechanics, 2003
This paper is a short essay on discontinuous Galerkin methods intended for a very wide audience. We present the discontinuous Galerkin methods and describe and discuss their main features. Since the methods use completely discontinuous approxima-tions, they produce mass matrices that are block-diagonal. This renders the methods highly parallelizable when applied to hyperbolic problems. Another consequence of the use of discontinuous approximations is that these methods can easily handle irregular meshes with hanging nodes and approximations that have polynomials of different degrees in different elements. They are thus ideal for use with adaptive algorithms. Moreover, the methods are locally conservative (a property highly valued by the computational fluid dynamics community) and, in spite of providing discontinuous approximations, stable, and high-order accurate. Even more, when applied to non-linear hyperbolic problems, the discontinuous Galerkin methods are able to capture highly complex solutions presenting discontinuities with high resolution. In this paper, we concentrate on the exposition of the ideas behind the devising of these methods as well as on the mechanisms that allow them to perform so well in such a variety of problems.
A high-order Discontinuous Galerkin method with Lagrange Multipliers (DGLM) is presented for the solution of advection–diffusion problems on unstructured adaptive meshes. Unlike other hybrid discretization methods for transport problems, it operates directly on the second-order form the advection–diffusion equation. Like the Discontinuous Enrichment Method (DEM), it chooses the basis functions among the free-space solutions of the homogeneous form of the governing partial differential equation, and relies on Lagrange multipliers for enforcing a weak continuity of the approximated solution across the element interface boundaries. However unlike DEM, the proposed hybrid discontinuous Galerkin method approximates the Lagrange multipliers in a space of polynomials, instead of a space of traces on the element boundaries of the normal derivatives of a subset of the basis functions. For a homogeneous problem, the design of arbitrarily high-order elements based on this DGLM method is supported by a detailed mathematical analysis. For a non-homogeneous one, the approximated solution is locally decomposed into its homogeneous and particular parts. The homogeneous part is captured by the DGLM elements designed for a homogenous problem. The particular part is obtained analytically after the source term is projected onto an appropriate polynomial space. This decoupling between the two parts of the solution is another differentiator between DGLM and DEM with attractive computational advantages. An a posteriori error estimator for the proposed method is also derived to enable adaptive mesh refinement. All theoretical results are illustrated by numerical simulations that furthermore highlight the potential of the proposed high-order hybrid DG method for transport problems with steep gradients in the high Péclet number regime.
High Order Discontinuous Galerkin Method
2004
Standard continuous Galerkin-based finite element methods have poor stability prop-erties when applied to transport-dominated flow problems, so excessive numerical sta-bilization is needed. In contrast, the Discontinuous Galerkin method is known to have good stability ...
Discontinuous Galerkin Methods for elliptic problems
1998
This paper is a short essay on discontinuous Galerkin methods intended for a very wide audience. We present the discontinuous Galerkin methods and describe and discuss their main features. Since the methods use completely discontinuous approximations, they produce mass matrices that are block-diagonal. This renders the methods highly parallelizable when applied to hyperbolic problems. Another consequence of the use of discontinuous approximations is that these methods can easily handle irregular meshes with hanging nodes and approximations that have polynomials of different degrees in different elements. They are thus ideal for use with adaptive algorithms. Moreover, the methods are locally conservative (a property highly valued by the computational fluid dynamics community) and, in spite of providing discontinuous approximations, stable, and high-order accurate. Even more, when applied to non-linear hyperbolic problems, the discontinuous Galerkin methods are able to capture highly complex solutions presenting discontinuities with high resolution. In this paper, we concentrate on the exposition of the ideas behind the devising of these methods as well as on the mechanisms that allow them to perform so well in such a variety of problems.
Journal of Computational Science, 2013
We revisit the finite element analysis of convection dominated flow problems within the recently developed Discontinuous Petrov-Galerkin (DPG) variational framework. We demonstrate how test function spaces that guarantee numerical stability can be computed automatically with respect to the so called optimal test space norm by using an element subgrid discretization. This should make the DPG method not only stable but also robust, that is, uniformly stable with respect to the Péclet number in the current application. The effectiveness of the algorithm is demonstrated on two problems for the linear advection-diffusion equation.
An introduction to the discontinuous Galerkin method for convection-dominated problems
Advanced numerical approximation of nonlinear …, 1998
In these notes, we study the Runge Kutta Discontinuous Galerkin method for numericaly solving nonlinear hyperbolic systems and its extension for convectiondominated problems, the so-called Local Discontinuous Galerkin method. Examples of problems to which these methods can be applied are the Euler equations of gas dynamics, the shallow water equations, the equations of magneto-hydrodynamics, the compressible Navier-Stokes equations with high Reynolds numbers, and the equations of the hydrodynamic model for semiconductor device simulation. The main features that make the methods under consideration attractive are their formal highorder accuracy, their nonlinear stability, their high parallelizability, their ability to handle complicated geometries, and their ability to capture the discontinuities or strong gradients of the exact solution without producing spurious oscillations. The purpose of these notes is to provide a short introduction to the devising and analysis of these discontinuous Galerkin methods. Aknowledgements. The author is grateful to Al o Quarteroni for the invitation to give a series of lectures at the CIME, June 23{28, 1997, the material of which is contained in these notes. He also thanks F. Bassi and F. Rebay, and I. Lomtev and G.E. Karniadakis for kindly providing pictures from their papers 2] and 3], and 46] and 65], respectively. 1 2 Contents Preface Chapter 1. A historical overview 1.1. The original Discontinuous Galerkin method 1.2. Nonlinear hyperbolic systems: The RKDG method 1.3. Convection-di usion systems: The LDG method 1.4. The content of these notes Chapter 2. The scalar conservation law in one space dimension 2.1. Introduction 2.2. The discontinuous Galerkin-space discretization 2.3. The TVD-Runge-Kutta time discretization 2.4. The generalized slope limiter 2.5. Computational results 2.6. Concluding remarks 2.7. Appendix: Proof of the L 2-error estimates in the linear case Chapter 3. The RKDG method for multidimensional systems 3.