An hp-Analysis of the Local Discontinuous Galerkin Method for Diffusion Problems

We present an hp-error analysis of the local discontinuous Galerkin method for diffusion problems, considering unstructured meshes with hanging nodes and two- and three-dimensional domains. Our estimates are optimal in the meshsize h and slightly suboptimal in the polynomial approximation order p. Optimality in p is achieved for matching grids and polynomial boundary conditions.

[1]  Clint Dawson,et al.  Some Extensions Of The Local Discontinuous Galerkin Method For Convection-Diffusion Equations In Mul , 1999 .

[2]  Ivo Babuška,et al.  The h-p version of the finite element method , 1986 .

[3]  E. Süli,et al.  Discontinuous hp-finite element methods for advection-diffusion problems , 2000 .

[4]  D. Arnold.  An Interior Penalty Finite Element Method with Discontinuous Elements , 1982 .

[5]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[6]  Bernardo Cockburn,et al.  Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems , 2002, Math. Comput..

[7]  PAUL CASTILLO,et al.  Performance of Discontinuous Galerkin Methods for Elliptic PDEs , 2002, SIAM J. Sci. Comput..

[8]  Ilaria Perugia,et al.  An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems , 2000, SIAM J. Numer. Anal..

[9]  Ilaria Perugia,et al.  The hp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations , 2003, Math. Comput..

[10]  B. Rivière,et al.  Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I , 1999 .

[11]  Endre Süli,et al.  hp-DGFEM on Shape-Irregular Meshes: Reaction-Diffusion Problems , 2001 .

[12]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..