Results 11  20
of
101
Adjoint consistency analysis of discontinuous Galerkin discretizations
 SIAM J. Numer. Anal
"... Abstract. This paper is concerned with the adjoint consistency of discontinuous Galerkin (DG) discretizations. Adjoint consistency—in addition to consistency—is the key requirement for DG discretizations to be of optimal order in L2 as well as measured in terms of target functionals. We provide a ge ..."
Abstract

Cited by 27 (5 self)
 Add to MetaCart
(Show Context)
Abstract. This paper is concerned with the adjoint consistency of discontinuous Galerkin (DG) discretizations. Adjoint consistency—in addition to consistency—is the key requirement for DG discretizations to be of optimal order in L2 as well as measured in terms of target functionals. We provide a general framework for analyzing the adjoint consistency of DG discretizations which is also useful for the derivation of adjoint consistent methods. This analysis will be performed for the DG discretizations of the linear advection equation, the interior penalty DG method for elliptic problems, and the DG discretization of the compressible Euler equations. This framework is then used to derive an adjoint consistent DG discretization of the compressible Navier–Stokes equations. Numerical experiments demonstrate the link of adjoint consistency to the accuracy of numerical flow solutions and the smoothness of discrete adjoint solutions.
Stabilized interior penalty methods for the timeharmonic Maxwell equations
 ComputerMethods in AppliedMechanics and Engineering
, 2002
"... We propose stabilized interior penalty discontinuous Galerkin methods for the indefinite time–harmonic Maxwell system. The methods are based on a mixed formulation of the boundary value problem chosen to provide control on the divergence of the electric field. We prove optimal error estimates for th ..."
Abstract

Cited by 26 (6 self)
 Add to MetaCart
We propose stabilized interior penalty discontinuous Galerkin methods for the indefinite time–harmonic Maxwell system. The methods are based on a mixed formulation of the boundary value problem chosen to provide control on the divergence of the electric field. We prove optimal error estimates for the methods in the special case of smooth coefficients and perfectly conducting boundary using a duality approach. Key words: Finite elements, discontinuous Galerkin methods, interior penalty methods, timeharmonic Maxwell’s equations 1
The hplocal discontinuous Galerkin method for lowfrequency timeharmonic Maxwell equations
 MATH. COMP
, 2001
"... The local discontinuous Galerkin method for the numerical approximation of the timeharmonic Maxwell equations in a lowfrequency regime is introduced and analyzed. Topologically nontrivial domains and heterogeneous media are considered, containing both conducting and insulating materials. The prese ..."
Abstract

Cited by 23 (9 self)
 Add to MetaCart
(Show Context)
The local discontinuous Galerkin method for the numerical approximation of the timeharmonic Maxwell equations in a lowfrequency regime is introduced and analyzed. Topologically nontrivial domains and heterogeneous media are considered, containing both conducting and insulating materials. The presented method involves discontinuous Galerkin discretizations of the curlcurl and graddiv operators, derived by introducing suitable auxiliary variables and socalled numerical fluxes. An hpanalysis is carried out and error estimates that are optimal in the meshsize h and slightly suboptimal in the approximation degree p are obtained.
Optimal BV estimates for a discontinuous Galerkin method in linear elasticity
 Applied Mathematics Research Express
"... Discontinuous Galerkin (DG) finiteelement methods for second and fourthorder elliptic problems were introduced about three decades ago. These methods stem from the hybrid methods developed by Pian and his coworker [25]. At the time of their introduc ..."
Abstract

Cited by 23 (6 self)
 Add to MetaCart
(Show Context)
Discontinuous Galerkin (DG) finiteelement methods for second and fourthorder elliptic problems were introduced about three decades ago. These methods stem from the hybrid methods developed by Pian and his coworker [25]. At the time of their introduc
Quasioptimal convergence rate of an adaptive discontinuous Galerkin method
 SIAM J. NUMER. ANAL
, 2010
"... We analyze an adaptive discontinuous finite element method (ADFEM) for symmetric second order linear elliptic operators. The method is formulated on nonconforming meshes made of simplices or quadrilaterals, with any polynomial degree and in any dimension ≥ 2. We prove that the ADFEM is a contraction ..."
Abstract

Cited by 21 (4 self)
 Add to MetaCart
We analyze an adaptive discontinuous finite element method (ADFEM) for symmetric second order linear elliptic operators. The method is formulated on nonconforming meshes made of simplices or quadrilaterals, with any polynomial degree and in any dimension ≥ 2. We prove that the ADFEM is a contraction for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops. We design a refinement procedure that maintains the level of nonconformity uniformly bounded, and prove that the approximation classes using continuous and discontinuous finite elements are equivalent. The geometric decay and the equivalence of classes are instrumental to derive optimal cardinality of ADFEM. We show that ADFEM (and AFEM on nonconforming meshes) yields a decay rate of energy error plus oscillation in terms of number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.
hpDISCONTINUOUS GALERKIN METHODS FOR THE HELMHOLTZ EQUATION WITH LARGE WAVE NUMBER
, 2009
"... This paper develops some interior penalty hpdiscontinuous Galerkin (hpDG) methods for the Helmholtz equation in two and three dimensions. The proposed hpDG methods are defined using a sesquilinear form which is not only meshdependent but also degreedependent. In addition, the sesquilinear form ..."
Abstract

Cited by 20 (3 self)
 Add to MetaCart
(Show Context)
This paper develops some interior penalty hpdiscontinuous Galerkin (hpDG) methods for the Helmholtz equation in two and three dimensions. The proposed hpDG methods are defined using a sesquilinear form which is not only meshdependent but also degreedependent. In addition, the sesquilinear form contains penalty terms which not only penalize the jumps of the function values across the element edges but also the jumps of the first order tangential derivatives as well as jumps of all normal derivatives up to order p. Furthermore, to ensure the stability, the penalty parameters are taken as complex numbers with positive imaginary parts. It is proved that the proposed hpdiscontinuous Galerkin methods are absolutely stable (hence, wellposed). For each fixed wave number k, suboptimal order error estimates in the broken H 1norm and the L 2norm are derived without any mesh constraint. The error estimates and the stability estimates are improved to optimal order under the mesh condition k 3 h 2 p −1 ≤ C0 by utilizing these stability and error estimates and using a stabilityerror iterative procedure To overcome the difficulty caused by strong indefiniteness of the Helmholtz problems in the stability analysis for numerical solutions, our main ideas for stability analysis are to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in [19, 20, 33], which enable us to derive stability estimates and error bounds with explicit dependence on the mesh size h, the polynomial degree p, the wave number k, as well as all the penalty parameters for the numerical solutions.
Review of A Priori Error Estimation for Discontinuous Galerkin Methods
, 2000
"... this report, all partitions P h are assumed to be shaperegular. In addition, we shall associate with each element K the element boundary @K. The unit normal vector outward from K (resp. K i ) is denoted by n (resp. nj i ). Given a partition P h , we shall denote the collection of edges of P h (poin ..."
Abstract

Cited by 20 (1 self)
 Add to MetaCart
(Show Context)
this report, all partitions P h are assumed to be shaperegular. In addition, we shall associate with each element K the element boundary @K. The unit normal vector outward from K (resp. K i ) is denoted by n (resp. nj i ). Given a partition P h , we shall denote the collection of edges of P h (points in one dimension, faces in three dimensions) by the set E h = f l g, l = 1; : : : ; N . Edges represent here open subsets of either W or @W. We thus introduce the set G int of interior edges as: G int =
The direct discontinuous Galerkin (ddg) methods for diffusion problems
 SIAM J. Numer. Anal
"... Abstract. Based on a novel numerical flux involving jumps of even order derivatives of the numerical solution, a direct discontinuous Galerkin (DDG) method for diffusion problems was introduced in [H. Liu and J. Yan, SIAM J. Numer. Anal. 47(1) (2009), 475698]. In this work, we show that higher orde ..."
Abstract

Cited by 20 (5 self)
 Add to MetaCart
(Show Context)
Abstract. Based on a novel numerical flux involving jumps of even order derivatives of the numerical solution, a direct discontinuous Galerkin (DDG) method for diffusion problems was introduced in [H. Liu and J. Yan, SIAM J. Numer. Anal. 47(1) (2009), 475698]. In this work, we show that higher order (k≥4) derivatives in the numerical flux can be avoided if some interface corrections are included in the weak formulation of the DDG method; still the jump of 2nd order derivatives is shown to be important for the method to be efficient with a fixed penalty parameter for all pk elements. The refined DDG method with such numerical fluxes enjoys the optimal (k+1)th order of accuracy. The developed method is also extended to solve convection diffusion problems in both one and twodimensional settings. A series of numerical tests are presented to demonstrate the high order accuracy of the method.
The local discontinuous Galerkin method for the Oseen equations
 Math. Comp
, 2002
"... We introduce and analyze the local discontinuous Galerkin method for the Oseen equations of incompressible fluid flow. For a class of shaperegular meshes with hanging nodes, we derive optimal a priori estimates for the errors in the velocity and the pressure in L 2 and negativeorder norms. Nume ..."
Abstract

Cited by 17 (5 self)
 Add to MetaCart
(Show Context)
We introduce and analyze the local discontinuous Galerkin method for the Oseen equations of incompressible fluid flow. For a class of shaperegular meshes with hanging nodes, we derive optimal a priori estimates for the errors in the velocity and the pressure in L 2 and negativeorder norms. Numerical experiments are presented which verify these theoretical results and show that the method performs well for a wide range of Reynolds numbers.
DISCONTINUOUS GALERKIN FINITE ELEMENT APPROXIMATION OF NONLINEAR SECONDORDER ELLIPTIC AND HYPERBOLIC SYSTEMS
"... Abstract. We develop the convergence analysis of discontinuous Galerkin finite element approximations to symmetric secondorder quasilinear elliptic and hyperbolic systems of partial differential equations in divergence form in a bounded spatial domain in R d, subject to mixed Dirichlet–Neumann boun ..."
Abstract

Cited by 16 (7 self)
 Add to MetaCart
(Show Context)
Abstract. We develop the convergence analysis of discontinuous Galerkin finite element approximations to symmetric secondorder quasilinear elliptic and hyperbolic systems of partial differential equations in divergence form in a bounded spatial domain in R d, subject to mixed Dirichlet–Neumann boundary conditions. Optimalorder asymptotic bounds are derived on the discretization error in each case without requiring the global Lipschitz continuity or uniform monotonicity of the stress tensor. Instead, only local smoothness and a G˚arding inequality are used in the analysis. Key words. Nonlinear elliptic and hyperbolic systems of partial differential equations, discontinuous Galerkin methods, Legendre–Hadamard condition, broken G˚arding inequality 1. Introduction. Secondorder nonlinear elliptic and hyperbolic systems of partial differential equations arise in numerous applications, and a substantial body of research has been devoted to their analytical and computational study. This paper is concerned with the construction and convergence analysis of a class of numerical algorithms — discontinuous Galerkin finite element methods — for the approximate solution of quasilinear elliptic and hyperbolic systems. Nonlinear elasticity is a