463 research outputs found
Robust Numerical Methods for Singularly Perturbed Differential Equations--Supplements
The second edition of the book "Roos, Stynes, Tobiska -- Robust Numerical
Methods for Singularly Perturbed Differential Equations" appeared many years
ago and was for many years a reliable guide into the world of numerical methods
for singularly perturbed problems. Since then many new results came into the
game, we present some selected ones and the related sources.Comment: arXiv admin note: text overlap with arXiv:1909.0827
Uniform convergence of optimal order under a balanced norm of a local discontinuous Galerkin method on a Shishkin mesh
For singularly perturbed reaction-diffusion problems in 1D and 2D, we study a
local discontinuous Galerkin (LDG) method on a Shishkin mesh. In these cases,
the standard energy norm is too weak to capture adequately the behavior of the
boundary layers that appear in the solutions. To deal with this deficiency, we
introduce a balanced norm stronger than the energy norm. In order to achieve
optimal convergence under the balanced norm in one-dimensional case, we design
novel numerical fluxes and propose a special interpolation that consists of a
Gauss-Radau projection and a local projection. Moreover, we generalize
the numerical fluxes and interpolation, and extend convergence analysis of
optimal order from 1D to 2D. Finally, numerical experiments are presented to
confirm the theoretical results.Comment: 22pages, two table
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