Local error analysis of the interior penalty discontinuous Galerkin method for second order elliptic problems

Abstract A local a priori and a posteriori analysis is developed for the Galerkin method with discontinuous finite elements for solving stationary diffusion problems. The main results are an optimalorder estimate for the point-wise error and a corresponding a posteriori error bound. The proofs are based on weighted L 2-norm error estimates for discrete Green functions as already known for the ‘continuous’ finite element method.

[1]  P. Hansbo,et al.  A finite element method for domain decomposition with non-matching grids , 2003 .

[2]  Guido Kanschat,et al.  A multilevel discontinuous Galerkin method , 2003, Numerische Mathematik.

[3]  Rolf Rannacher,et al.  Some Optimal Error Estimates for Piecewise Linear Finite Element Approximations , 1982 .

[4]  D. Arnold,et al.  Discontinuous Galerkin Methods for Elliptic Problems , 2000 .

[5]  Rolf Rannacher,et al.  An Optimal Control Approach to A-Posteriori Error Estimation , 2001 .

[6]  R. Rannacher,et al.  Asymptotic error expansion and Richardson extranpolation for linear finite elements , 1986 .

[7]  Rolf Rannacher,et al.  Finite element methods for nonlinear elliptic systems of second order , 1980 .

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

[9]  Rolf Rannacher,et al.  A Feed-Back Approach to Error Control in Finite Element Methods: Basic Analysis and Examples , 1996 .

[10]  Wolfgang Bangerth,et al.  Concepts for Object-Oriented Finite Element Software - the deal.II Library , 1999 .

[11]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[12]  Peter Hansbo,et al.  Energy norm a posteriori error estimation for discontinuous Galerkin methods , 2003 .

[13]  J. Nitsche.  Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind , 1971 .

[14]  Serge Nicaise,et al.  Nitsche mortar finite element method for the transmission problem with singularities , 2001 .

[15]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[16]  P. Grisvard.  Singularities in Boundary Value Problems , 1992 .

[17]  Rolf Rannacher,et al.  An optimal control approach to a posteriori error estimation in finite element methods , 2001, Acta Numerica.

[18]  A Review of A Posteriori Error Estimation , 1996 .