Justification of the saturation assumption

• Verfasst von: Carstensen, Carsten [VerfasserIn]
• Verfasst von: Gallistl, Dietmar [VerfasserIn]
• Verfasst von: Gedicke, Joscha [VerfasserIn]
• Titel: Justification of the saturation assumption
• Verf.angabe: C. Carstensen, D. Gallistl and J. Gedicke
• Jahr: 2016
• Jahr des Originals: 2015
• Umfang: 25 S.
• Fussnoten: Published 28 October 2015 ; Gesehen am 29.04.2020
• Titel Quelle: Enthalten in: Numerische Mathematik
• Ort Quelle: Berlin : Springer, 1959
• Jahr Quelle: 2016
• Band/Heft Quelle: 134(2016), 1, Seite 1-25
• ISSN Quelle: 0945-3245
• DOI: doi:10.1007/s00211-015-0769-7
• Datenträger: Online-Ressource
• Sprache: eng
• K10plus-PPN: 1696822351

Abstract

The saturation assumption is widely used in computational science and engineering, usually without any rigorous theoretical justification and even despite of counterexamples for some coarse meshes known in the mathematical literature. On the other hand, there is overwhelming numerical evidence at least in an asymptotic regime for the validity of the saturation. In particular, the strong saturation assumption holds for all triangulations with more than one degree of freedom. The weak saturation test (WS) is only required for zero or one degree of freedom and gives a definite outcome with O(1) operations. The only counterexamples known so far are regular n-polygons. The paper also discusses a generalization to linear elliptic second-order PDEs with small convection to prove that saturation is somehow generic and fails only in very particular situations characterised by (WS).