NA-06 / 15 A-posteriori error analysis of hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic problems

We develop the a-posteriori error analysis of hp-version interior-penalty discontinuous Galerkin finite element methods for a class of second-order quasilinear elliptic partial differential equations. Computable upper and lower bounds on the error are derived in terms of a natural (mesh-dependent) energy norm. The bounds are explicit in the local mesh size and the local degree of the approximating polynomial. The performance of the proposed estimators within an automatic hp-adaptive refinement procedure is studied through numerical experiments.

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

[2]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[3]  P. Raviart,et al.  On a Finite Element Method for Solving the Neutron Transport Equation , 1974 .

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

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

[6]  Jindřich Nečas,et al.  Introduction to the Theory of Nonlinear Elliptic Equations , 1986 .

[7]  Juhani Pitkäranta,et al.  An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation , 1986 .

[8]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[9]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[10]  John W. Barrett,et al.  Quasi-norm error bounds for the finite element approximation of some degenerate quasilinear elliptic equations and variational inequalities , 1994 .

[11]  Chi-Wang Shu,et al.  The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V , 1998 .

[12]  I. Babuska,et al.  A DiscontinuoushpFinite Element Method for Diffusion Problems , 1998 .

[13]  PAUL HOUSTON,et al.  Stabilized hp-Finite Element Methods for First-Order Hyperbolic Problems , 2000, SIAM J. Numer. Anal..

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

[15]  Joseph E. Flaherty,et al.  Software for the parallel adaptive solution of conservation laws by discontinous Galerkin methods. , 2000 .

[16]  Christine Bernardi,et al.  An error indicator for mortar element solutions to the Stokes problem , 2001 .

[17]  Barbara I. Wohlmuth,et al.  On residual-based a posteriori error estimation in hp-FEM , 2001, Adv. Comput. Math..

[18]  Ilaria Perugia,et al.  Superconvergence of the Local Discontinuous Galerkin Method for Elliptic Problems on Cartesian Grids , 2001, SIAM J. Numer. Anal..

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

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

[21]  Endre Süli,et al.  Sobolev Regularity Estimation for hp-Adaptive Finite Element Methods , 2003 .

[22]  Jens Markus Melenk,et al.  HP-INTERPOLATION OF NON-SMOOTH FUNCTIONS , 2003 .

[23]  P. Frauenfelder,et al.  Exponential convergence of the hp-DGFEM for diffusion problems , 2003 .

[24]  D. Schötzau,et al.  Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Stokes problem , 2003 .

[25]  Endre Süli,et al.  Adaptive Finite Element Approximation of Hyperbolic Problems , 2003 .

[26]  Ilaria Perugia,et al.  Mixed discontinuous Galerkin approximation of the Maxwell operator: The indefinite case , 2005 .

[27]  G. Gatica,et al.  A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. Part I: a priori error analysis , 2004 .

[28]  Gabriel N. Gatica,et al.  A Local Discontinuous Galerkin Method for Nonlinear Diffusion Problems with Mixed Boundary Conditions , 2004, SIAM J. Sci. Comput..

[29]  E. Süli,et al.  Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: the scalar case , 2005 .

[30]  E. Süli,et al.  A note on the design of hp-adaptive finite element methods for elliptic partial differential equations , 2005 .

[31]  D. Schötzau,et al.  An hp-adaptive mixed discontinuous Galerkin FEM for nearly incompressible linear elasticity , 2006 .

[32]  J. Melenk,et al.  An adaptive strategy for hp-FEM based on testing for analyticity , 2007 .

[33]  Gary R. Consolazio,et al.  Finite Elements , 2007, Handbook of Dynamic System Modeling.

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

[35]  Endre Süli,et al.  Discontinuous Galerkin Finite Element Approximation of Nonlinear Second-Order Elliptic and Hyperbolic Systems , 2007, SIAM J. Numer. Anal..