A NUMERICAL STUDY OF UNIFORM SUPERCONVERGENCE OF LDG METHOD FOR SOLVING SINGULARLY PERTURBED PROBLEMS

In this paper, we consider the local discontinuous Galerkin method (LDG) for solving singularly perturbed convection-diffusion problems in one- and two-dimensional settings. The existence and uniqueness of the LDG solutions are verified. Numerical experiments demonstrate that it seems impossible to obtain uniform superconvergence for numerical fluxes under uniform meshes. Thanks to the implementation of two-type different anisotropic meshes, i.e., the Shishkin and an improved grade meshes, the uniform 2p + 1-order superconvergence is observed numerically for both one-dimensional and twodimensional cases.

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

[2]  L. Wahlbin,et al.  On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions , 1983 .

[3]  Hans-Görg Roos,et al.  A comparison of the finite element method on Shishkin and Gartland-type meshes for convection-diffusion problems , 1997 .

[4]  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..

[5]  J. Oden,et al.  A discontinuous hp finite element method for convection—diffusion problems , 1999 .

[6]  Christoph Schwab,et al.  The p and hp versions of the finite element method for problems with boundary layers , 1996, Math. Comput..

[7]  Paul Castillo,et al.  An Optimal Estimate for the Local Discontinuous Galerkin Method , 2000 .

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

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

[10]  Shishkin,et al.  A Finite Difference Scheme on a Priori Adapted Meshes for a Singularly Perturbed Parabolic Convection-Diffusion Equation , 2008 .

[11]  Ionel M. Navon,et al.  Mesh refinement strategies for solving singularly perturbed reaction-diffusion problems☆☆☆ , 2001 .

[12]  Chi-Wang Shu,et al.  Discontinuous Galerkin Methods: Theory, Computation and Applications , 2011 .

[13]  S. Rebay,et al.  A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations , 1997 .

[14]  George Em Karniadakis,et al.  The Development of Discontinuous Galerkin Methods , 2000 .

[15]  Zhimin Zhang,et al.  SUPERCONVERGENCE OF DG METHOD FOR ONE-DIMENSIONAL SINGULARLY PERTURBED PROBLEMS , 2007 .

[16]  L. Wahlbin,et al.  Local behavior in finite element methods , 1991 .

[17]  Jeanne Stynes,et al.  A Parameter-Uniform Finite Difference Method for a Coupled System of Convection-Diffusion Two-Point Boundary Value Problems , 2008 .

[18]  Fatih Celiker,et al.  Superconvergence of the numerical traces of discontinuous Galerkin and Hybridized methods for convection-diffusion problems in one space dimension , 2007, Math. Comput..

[19]  Chen,et al.  Uniform Convergence Analysis for Singularly Perturbed Elliptic Problems with Parabolic Layers , 2008 .

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

[21]  Lutz Tobiska,et al.  Numerical Methods for Singularly Perturbed Differential Equations , 1996 .

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

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

[24]  Hans-Görg Roos,et al.  Sufficient Conditions for Uniform Convergence on Layer-Adapted Grids , 1999, Computing.

[25]  Kenneth Eriksson,et al.  Adaptive finite element methods for parabolic problems II: optimal error estimates in L ∞ L 2 and L ∞ L ∞ , 1995 .

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

[27]  Kenneth Eriksson,et al.  Adaptive finite element methods for parabolic problems. I.: a linear model problem , 1991 .

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

[29]  Torsten Linß,et al.  Numerical methods on Shishkin meshes for linear convection-diffusion problems , 2001 .

[30]  J. Oden,et al.  hp-Version discontinuous Galerkin methods for hyperbolic conservation laws , 1996 .

[31]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems , 1989 .

[32]  Christos Xenophontos,et al.  THE hp FINITE ELEMENT METHOD FOR SINGULARLY PERTURBED PROBLEMS IN SMOOTH DOMAINS , 1998 .

[33]  Zhimin Zhang,et al.  Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems , 2003, Math. Comput..