Abstract
A difference method is presented for singularly perturbed convection-diffusion problems with discretization error estimates of high order (orderp), which hold uniformly in the singular perturbation parameterε. The method is based on the use of a defect-correction technique and special adaptively graded and patched meshes, with meshsizes varying betweenh andε 3/2 h whenp=2, whereh is the meshsize, used in the part of the domain where the solution is smooth, andε 3/2 h is the final meshsize in the boundary layer. Similar constructions hold for interior layers. The correction operator is a monotone operator, enabling the estimate of the error of optimal order in maximum norm. The total number of meshpoints used in ad-dimensional problem isO(ε −s)h −d+O(h −d), wheres is 1/p or 1/2p, respectively in the case of boundary or interior layer.
Zusammenfassung
Für singulär gestörte Konvektions-Diffusions-Probleme wird ein Differenzenverfahren mit Verfahrenfehlerschätzungen hoher Ordnung (Ordnungp) vorgestellt, die gleichmäßig im Parameter der singulären Störung sind. Das Verfahren beruht auf einer Defektkorrektur-Technik und auf speziellen adaptiv geteilten und gestückelten Gittern, deren Gitterweiten im Fallp=2 zwischenh undhε 3/2 variieren; dabei isth die Weite in den Bereichen mit glatter Lösung undhε 3/2 die Endweite in der Grenzschicht. Für innere Schichten läßt sich die Konstruktion analog durchführen. Der Korrektur-Operator ist monoton, was eine optimale Fehlerschätzung im Sinne der Maximumnorm gestattet. Die Gesamtzahl von Gitterpunkten bei einemd-dimensionalen Problem beträgtO(h −d)+h −d O(ε −s), mits=1/p bzw. 1/2p für Randbzw. Innenschichten.
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Axelsson, O., Nikolova, M. Adaptive refinement for convection-diffusion problems based on a defect-correction technique and finite difference method. Computing 58, 1–30 (1997). https://doi.org/10.1007/BF02684469
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DOI: https://doi.org/10.1007/BF02684469