Interior penalty discontinuous approximations of convection–diffusion problems with parabolic layers

SummaryA nonsymmetric discontinuous Galerkin finite element method with interior penalties is considered for two–dimensional convection–diffusion problems with regular and parabolic layers. On an anisotropic Shishkin–type mesh with bilinear elements we prove error estimates (uniformly in the perturbation parameter) in an integral norm associated with this method. On different types of interelement edges we derive the values of discontinuity–penalization parameters. Numerical experiments complement the theoretical results.

[1]  Hans-Görg Roos,et al.  A Priori Estimates for the Solution of Convection-Diffusion Problems and Interpolation on Shishkin Meshes , 1997 .

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

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

[4]  Torsten Linß,et al.  Layer-adapted meshes for convection-diffusion problems , 2003 .

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

[6]  Paul Houston,et al.  Discontinuous hp-Finite Element Methods for Advection-Diffusion-Reaction Problems , 2001, SIAM J. Numer. Anal..

[7]  Hans-Görg Roos,et al.  The Discontinuous Galerkin Finite Element Method for Singularly Perturbed Problems , 2003 .

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

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

[10]  Torsten Linß.  Anisotropic meshes and streamline-diffusion stabilization for convection-diffusion problems , 2005 .

[11]  Hans-Görg Roos.  Optimal Convergence of Basic Schemes for Elliptic Boundary Value Problems with Strong Parabolic Layers , 2002 .

[12]  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..