Journal ArticleDOI
Layer‐Adapted Grids for Singular Perturbation Problems
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In this paper, a survey on the application of Shishkin grids to convection-diffusion problems with dominant convection is given, further some new results and open problems are presented.Abstract:
In the present paper a survey is given on the application of Shishkin grids to convection-diffusion problems with dominant convection, further some new results and open problems are presented. The practical importance of these simplestructured grids lies in the possibility to resolve layers - the alternative technique of exponential fitting is not always successful. The use of uniformly stable method on a carefully chosen Shishkin mesh often leads to a uniformly convergent method.read more
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Journal ArticleDOI
Sufficient conditions for uniform convergence on layer-adapted grids
Hans-Görg Roos,Torsten Linß +1 more
TL;DR: Conditions on the mesh-characterizing function are derived that are sufficient for the convergence of the method, uniformly with respect to the perturbation parameter, and enable one to immediately deduce the rate of convergence.
Journal ArticleDOI
A brief survey on numerical methods for solving singularly perturbed problems
Mohan K. Kadalbajoo,Vikas Gupta +1 more
TL;DR: This survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems.
Journal ArticleDOI
Layer-adapted meshes for convection-diffusion problems
TL;DR: T theoretical results are presented that demonstrate that the use of properly layer-adapted meshes yields robust methods, i.e., methods that perform equally well no matter how dominant the convection.
Journal ArticleDOI
The SDFEM for a Convection-Diffusion Problem with a Boundary Layer: Optimal Error Analysis and Enhancement of Accuracy
Martin Stynes,Lutz Tobiska +1 more
TL;DR: An error bound is proved for the streamline-diffusion finite element method, showing that $u-u^N$ is superclose to $u^I$, which allows the construction of a simple postprocessing that yields a more accurate solution.
Journal ArticleDOI
Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems
TL;DR: In this work, the bilinear finite element method on a Shishkin mesh for convection-diffusion problems is analyzed in the two-dimensional setting and an e-uniform convergence of order N-3/2ln5/2N + eN-1ln1/ 2N in the L∞ norm is proved for some mesh points in the boundary layer region.