Anisotropic mesh refinement for singularly perturbed reaction diffusion problems
The paper is concerned with the finite element resolution of layers appearing in singularly perturbed problems. A special anisotropic grid of Shishkin type is constructed for reaction diffusion problems. Estimates of the finite element error in the energy norm are derived for two methods, namely the...
| Main Authors: | , |
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| Language: | English |
| Published: |
Technische Universität Chemnitz
1998
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| Subjects: | |
| Online Access: | http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199801080 https://monarch.qucosa.de/id/qucosa%3A17483 https://monarch.qucosa.de/api/qucosa%3A17483/attachment/ATT-0/ https://monarch.qucosa.de/api/qucosa%3A17483/attachment/ATT-1/ https://monarch.qucosa.de/api/qucosa%3A17483/attachment/ATT-2/ |
| Summary: | The paper is concerned with the finite element resolution of layers appearing
in singularly perturbed problems. A special anisotropic grid of Shishkin type
is constructed for reaction diffusion problems. Estimates of the finite element
error in the energy norm are derived for two methods, namely the standard
Galerkin method and a stabilized Galerkin method. The estimates are uniformly
valid with respect to the (small) diffusion parameter. One ingredient is a
pointwise description of derivatives of the continuous solution. A numerical
example supports the result.
Another key ingredient for the error analysis is a refined estimate for
(higher) derivatives of the interpolation error. The assumptions on admissible
anisotropic finite elements are formulated in terms of geometrical conditions
for triangles and tetrahedra. The application of these estimates is not
restricted to the special problem considered in this paper. |
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