Discontinuous Galerkin Finite Element Method for the Wave Equation.
ABSTRACT The symmetric interior penalty discontinuous Galerkin nite element method is presented for the numerical discretization of the second-order scalar wave equation. The resulting stiness matrix is symmetric positive denite and the mass matrix is essentially diagonal; hence, the method is inherently parallel and leads to fully explicit time integration. Optimal a priori error bounds are derived in the energy norm and the L2-norm for the semi-discrete formulation. In particular, the error in the energy norm is shown to converge with the optimal orderO(hminfs;'g) with respect to the mesh size h, the polynomial degree ', and the regularity exponent s of the continuous solution. Under additional regularity assumptions, the L2-error is shown to converge with the optimal orderO(h'+1). Numerical results results conrm the expected convergence rates and illustrate the versatility of the method.
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ABSTRACT: An overview of the deal.II library is given. This library provides the functionality needed by modern numerical software used in the nite element solution of partial dierential equations, oering adaptively rened meshes, dierent nite element classes, multigrid solvers and support for one, two and three spatial dimensions. We give a description of the basic design criteria used in the development of the library and how they were transformed into actual code, and some examples of the use of the library in numerical analysis. 1 Design and evolution of deal.II The DEAL project, short for Dierential Equations Analysis Library, was started to provide means for the implementation of adaptive nite element methods. In fact, the development of DEAL and adaptive methods at the Institute of Applied Mathematics in Heidelberg are closely linked. From this starting point, a nite element library was needed, that is able to handle grids with strongly varying mesh width and supports stra...11/1999;
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ABSTRACT: 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.Journal of Scientific Computing 08/2001; 16(3):173-261. · 1.71 Impact Factor
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ABSTRACT: 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 dis- continuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems. Key words. elliptic problems, discontinuous Galerkin, interior penalty AMS subject classification. 65N30 PII. S0036142901384162