A new error analysis for discontinuous finite element methods for linear elliptic problems

The standard a priori error analysis of discontinuous Galerkin methods requires additional regularity on the solution of the elliptic boundary value problem in order to justify the Galerkin orthogonality and to handle the normal derivative on element interfaces that appear in the discrete energy norm. In this paper, a new error analysis of discontinuous Galerkin methods is developed using only the H k weak formulation of a boundary value problem of order 2k. This is accomplished by replacing the Galerkin orthogonality with estimates borrowed from a posteriori error analysis and by using a discrete energy norm that is well defined for functions in H k .

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

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

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

[4]  J. Douglas,et al.  Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods , 1976 .

[5]  R. B. Kellogg,et al.  SINGULARITIES IN INTERFACE PROBLEMS , 1971 .

[6]  D. Schötzau,et al.  ENERGY NORM A POSTERIORI ERROR ESTIMATION OF hp-ADAPTIVE DISCONTINUOUS GALERKIN METHODS FOR ELLIPTIC PROBLEMS , 2007 .

[7]  Paul Houston,et al.  An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems , 2011 .

[8]  P. Raviart,et al.  Conforming and nonconforming finite element methods for solving the stationary Stokes equations I , 1973 .

[9]  Susanne C. Brenner,et al.  A W-cycle algorithm for a weakly over-penalized interior penalty method , 2007 .

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

[11]  Susanne C. Brenner,et al.  Two-level additive Schwarz preconditioners for nonconforming finite element methods , 1996, Math. Comput..

[12]  Carsten Carstensen,et al.  A unifying theory of a posteriori error control for discontinuous Galerkin FEM , 2009, Numerische Mathematik.

[13]  Susanne C. Brenner,et al.  Convergence of nonconforming multigrid methods without full elliptic regularity , 1999, Math. Comput..

[14]  S. C. Brenner,et al.  Poincaré–Friedrichs Inequalities for Piecewise H 2 Functions , 2004 .

[15]  Xiaobing Feng,et al.  Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition , 2007, Math. Comput..

[16]  T. Hughes,et al.  Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity , 2002 .

[17]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[18]  Susanne C. Brenner,et al.  A Weakly Over-Penalized Non-Symmetric Interior Penalty Method , 2007 .

[19]  Mary F. Wheeler,et al.  A Priori Error Estimates for Finite Element Methods Based on Discontinuous Approximation Spaces for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[20]  Thirupathi Gudi,et al.  Mixed Discontinuous Galerkin Finite Element Method for the Biharmonic Equation , 2008, J. Sci. Comput..

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

[22]  Igor Mozolevski,et al.  hp-Version a priori Error Analysis of Interior Penalty Discontinuous Galerkin Finite Element Approximations to the Biharmonic Equation , 2007, J. Sci. Comput..

[23]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[24]  S. C. Brenner,et al.  POINCAR´ E-FRIEDRICHS INEQUALITIES FOR PIECEWISE H 1 FUNCTIONS ∗ , 2003 .

[25]  G. A. Baker.  Finite element methods for elliptic equations using nonconforming elements , 1977 .

[26]  R. Verfiirth.  A posteriori error estimation and adaptive mesh-refinement techniques , 2001 .

[27]  Susanne C. Brenner,et al.  C0 Interior Penalty Methods for Fourth Order Elliptic Boundary Value Problems on Polygonal Domains , 2005, J. Sci. Comput..

[28]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis: Oden/A Posteriori , 2000 .

[29]  S. C. Brenner,et al.  An a posteriori error estimator for a quadratic C0-interior penalty method for the biharmonic problem , 2010 .

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

[31]  Susanne C. Brenner.  Discrete Sobolev and Poincaré inequalities for piecewise polynomial functions. , 2004 .

[32]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[33]  Ohannes A. Karakashian,et al.  A Posteriori Error Estimates for a Discontinuous Galerkin Approximation of Second-Order Elliptic Problems , 2003, SIAM J. Numer. Anal..

[34]  Xiaobing Feng,et al.  Two-Level Non-Overlapping Schwarz Preconditioners for a Discontinuous Galerkin Approximation of the Biharmonic Equation , 2005, J. Sci. Comput..

[35]  M. Wheeler.  An Elliptic Collocation-Finite Element Method with Interior Penalties , 1978 .

[36]  Susanne C. Brenner,et al.  Poincaré-Friedrichs Inequalities for Piecewise H1 Functions , 2003, SIAM J. Numer. Anal..

[37]  Susanne C. Brenner,et al.  Two-Level Additive Schwarz Preconditioners for a Weakly Over-Penalized Symmetric Interior Penalty Method , 2011, J. Sci. Comput..