The discontinuous Galerkin finite element method for singularly perturbed problems. (English) Zbl 1043.65130

Roos, Hans-Görg; Zarin, Helena

Bänsch, Eberhard (ed.), Challenges in scientific computing – CISC 2002. Proceedings of the conference “challenges in scientific computing”, Berlin, Germany, October 2–5, 2002. Berlin: Springer (ISBN 3-540-40887-8/hbk). Lect. Notes Comput. Sci. Eng. 35, 246-267 (2003).

Summary: A nonsymmetric discontinuous Galerkin finite element method with interior penalties is considered for two-dimensional singularly perturbed problems. On an anisotropic Shishkin mesh with bilinear elements we prove error estimates (uniformly in the perturbation parameter) in an integral norm associated with this method. We perform separate analyses for the cases of reaction-diffusion and convection-diffusion problems. On different types of interelement edges we derive the values of discontinuity-penalization parameters. Numerical experiments support the theoretical results.
For the entire collection see [Zbl 1029.00037].

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25	Boundary value problems for second-order elliptic equations
35B25	Singular perturbations in context of PDEs
65N15	Error bounds for boundary value problems involving PDEs
65N50	Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs

Keywords:

reaction-diffusion equation; singular perturbation; nonsymmetric discontinuous Galerkin finite element; Shishkin mesh; error estimates; convection-diffusion problems; numerical experiments