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An analysis of discontinuous Galerkin methods for elliptic problems

Reinhold Schneider, Yuesheng Xu, Aihui Zhou
2006 Advances in Computational Mathematics  
Micchelli on the occasion of his 60th birthday with friendship and esteem We study global and local behaviors for three kinds of discontinuous Galerkin schemes for elliptic equations of second order.  ...  These theoretical results are applied to develop local/parallel and adaptive finite element methods, based on the discontinuous Galerkin methods.  ...  Introduction The discontinuous Galerkin methods for elliptic and parabolic equations were formerly called interior penalty methods, whose development remained independent of that of the discontinuous Galerkin  ... 
doi:10.1007/s10444-004-7619-y fatcat:jlkhv5bg6nax7ihsvzrx6qjgja

An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems

Paul Castillo, Bernardo Cockburn, Ilaria Perugia, Dominik Schötzau
2000 SIAM Journal on Numerical Analysis  
In this paper, we present the rst a priori error analysis for the Local Discontinuous Galerkin method for a model elliptic problem.  ...  For arbitrary meshes with hanging nodes and elements of various shapes, we show that, for stabilization parameters of order one, the L 2 -norm of the gradient and the L 2 -norm of the potential are of  ...  In this paper, we present the rst a priori error analysis of the Local Discontinuous Galerkin (LDG) method for the following classical model elliptic problem: ? u = f in ; u = g D on ?  ... 
doi:10.1137/s0036142900371003 fatcat:shm5i33rebarldzpb55iojg5ny

An Analysis of the Embedded Discontinuous Galerkin Method for Second-Order Elliptic Problems

Bernardo Cockburn, Johnny Guzmán, See-Chew Soon, Henry K. Stolarski
2009 SIAM Journal on Numerical Analysis  
In this paper, we consider embedded methods for second-order elliptic problems obtained from hybridizable discontinuous methods by changing the space of the hybrid unknown from discontinuous to continuous  ...  The embedded discontinuous Galerkin methods are obtained from hybridizable discontinuous Galerkin methods by a simple change of the space of the hybrid unknown.  ...  to E h of the approximation provided by the continuous Galerkin method.  ... 
doi:10.1137/080726914 fatcat:npzz5v72pfastd52n6i574hrjq

Theory and Applications of Discontinuous Galerkin Methods

Susanne Brenner, Ronald Hoppe, Béatrice Rivière
2012 Oberwolfach Reports  
Theory and Application of Discontinuous Galerkin Methods. Mathematics Subject Classification (2000): 65N30, 65N55, 65K15, 65Y10.  ...  Peter) Convergence Analysis of an Adaptive Interior Penalty Discontinuous Galerkin Method for the Biharmonic Problem . . . . . . . . . . . . . . . . . . . . . . . 569 Marcus J.  ...  These issues include (but are not limited to) the formulation of more efficient methods with fewer number of degrees of freedom, the formulation of methods for coupled problems, the analysis of guaranteed  ... 
doi:10.4171/owr/2012/10 fatcat:6dtrw4zbgfb2lctct2idxxshmq

An analysis of HDG methods for the Helmholtz equation

J. Cui, W. Zhang
2013 IMA Journal of Numerical Analysis  
Zhang, A posteriori error analysis for hybridizable discontinuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal., 51(1) 2013, 676-693. [3] B. Cockburn and W.  ...  Research Interest Numerical method for second order fully nonlinear elliptic PDEs Finite element method for liquid crystals A posteriori error estimation and adaptivity Discontinuous Galerkin method Publications  ...  Zhang, A posteriori error analysis for hybridizable discontinuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal., 51(1) 2013, 676-693.[3] B. Cockburn and W.  ... 
doi:10.1093/imanum/drt005 fatcat:qfufusiuxveejkxpprn4w7m524

Foreword

Bernardo Cockburn, Chi-Wang Shu
2009 Journal of Scientific Computing  
In [2], an a posteriori error analysis of an over-penalized symmetric interior penalty method is presented.  ...  In [16], a novel approach, namely, the weighted-residual formulation of the discontinuous Galerkin methods, is used to derive a-posteriori error estimates for all known discontinuous Galerkin methods in  ...  In [12] , the analysis of a fully discrete symmetric interior penalty discontinuous Galerkin method for the wave equation is presented.  ... 
doi:10.1007/s10915-009-9298-9 fatcat:3ykthqnr4ja2lnuwjpklmu2kxu

Page 2759 of Mathematical Reviews Vol. , Issue 2002D [page]

2002 Mathematical Reviews  
Galerkin methods for elliptic problems.  ...  Summary: “We provide a common framework for the understand- ing, comparison, and analysis of several discontinuous Galerkin methods that have been proposed for the numerical treatment of elliptic problems  ... 

Page 2722 of Mathematical Reviews Vol. , Issue 99d [page]

1991 Mathematical Reviews  
discontinuous in time for n-dimensional 65 NUMERICAL ANALYSIS 2722 initial-boundary value parabolic problems.  ...  (BR-ECPM; Campinas); Cortina, Elsa Fourier analysis of Petrov-Galerkin methods based on biorthogonal multiresolution analyses.  ... 

Page 1262 of Mathematical Reviews Vol. , Issue 2002B [page]

2002 Mathematical Reviews  
2002b:65005 Franco Brezzi, Bernardo Cockburn and Donatella Marini, Discon- tinuous Galerkin methods for elliptic problems (89-101); Richard S.  ...  Baaijens, Successes and failures of discontinuous Galerkin methods in viscoelastic fluid analysis (263-270).  ... 

Page 7246 of Mathematical Reviews Vol. , Issue 87m [page]

1987 Mathematical Reviews  
Summary: “We consider the generalized Galerkin method for the third boundary value problem of the second order elliptic equation —F(k$e) — F(k$e) + qu=fs, (z,y) EQ, (k2% + bu) |p =0.  ...  Lund, John (1-MTS) 87m:65186 Symmetrization of the sinc-Galerkin method for boundary value problems. Math. Comp. 47 (1986), no. 176, 571-588. The sinc-Galerkin method developed in a paper by F.  ... 

Page 5120 of Mathematical Reviews Vol. , Issue 97H [page]

1997 Mathematical Reviews  
“The method is based on the reduction of the Cauchy problem to a continuation problem. An integral equation method is used for the stable numerical solution of the derived continuation problem.  ...  (PRC-XJU-AM; Xi’an); Li, Kaitai (PRC-XJU-AM; Xi’an) Stability analysis of nonlinear Galerkin & Galerkin methods for nonlinear evolution equations.  ... 

Page 8135 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews  
F. (1-TX-MD; Austin, TX) A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems. Log number: R74.  ...  This article deals with a posteriori error estimates for a class of dis- continuous Galerkin methods applied to elliptic boundary value problems in two dimensions.  ... 

A Posteriori L_∞ (L_2 )+L_2 (H^1 )–Error Bounds in Discontinuous Galerkin Methods For Semidiscrete Semilinear Parabolic Interface Problems

2021 Baghdad Science Journal  
A key idea for this technique is the use of error estimators derived for elliptic interface problems to obtain parabolic estimators that are of optimal order in space and time.  ...  More specifically, optimal order a posteriori error analysis in the ∞ ( 2 ) + 2 ( 1 )norm for semidiscrete semilinear parabolic interface problems is derived by using elliptic reconstruction technique  ...  In Section 2, the model problem is introduced and discontinuous Galerkin method, with some necessary background results, are discussed.  ... 
doi:10.21123/bsj.2021.18.3.0522 fatcat:eqpwqocpsvd25n7lfdcdvjegfm

Page 3596 of Mathematical Reviews Vol. , Issue 2000e [page]

2000 Mathematical Reviews  
), the finite difference method (FDM), the Ritz-Galerkin method (RGM), etc., for solv- ing complicated problems of partial differential equations (PDE).  ...  Summary: “A method to estimate and control pointwise errors in finite element approximations of elliptic problems is presented as an application of the more general theory of goal-oriented error estimation  ... 

Page 2914 of Mathematical Reviews Vol. , Issue 90E [page]

1990 Mathematical Reviews  
They provide an outline of the analysis of the Galerkin method for the solution of the potential problem for the Laplacian in a bounded plane domain, subject to nonlinear boundary conditions refor- mulated  ...  finite element methods for second-order elliptic problems.  ... 
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