Abstract
Our goal is to develop adaptive strategies in order to obtain finite element solutions of the partial differential equation-Δu=f(u) in a bounded domain Ω ⊆ ℝ2. In practice one works with an approximationf h off. But this may give wrong results if we do not control the coresponding approximation error on coarse girds. In this work we develop a strategy that is robust, but less efficient, in the beginning of the adaptive algorithm and switches to a more efficient procedure if certainsaturation conditions are satisfied. The results are based on a posteriori saturation criterial that measure the quality of the approximation solution.
Zusammenfassung
Wir entwickeln adaptive Methoden zur Berechnung von Finite Elemente-Lösungen der Partiellen Differentialgleichung −Δu=f(u) auf einem beschränktem Gebiet Ω ⊆ ℝ2. In der Praxis arbeitet man mit einer Approximationf h vonf, was zu falschen Ergebnissen führen kann, wenn man den zugehörigen Approximationsfehler auf dem groben Gitter nicht mitberücksichtigt. Wir verwenden eine Strategie, die zu Beginn der Iteration robust aber weniger effizient ist und gehen zu effektiveren Methoden über, falls gewisse Sättigungsbedingungen erfühlt sind. Dazu leiten wir a posteriori Fehlerschranken und a posteriori Sättigungsbedingungen her, um die Qualität der numerischen Lösung zu beurteilen.
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Dörfler, W. A robust adaptive strategy for the nonlinear Poisson equation. Computing 55, 289–304 (1995). https://doi.org/10.1007/BF02238484
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DOI: https://doi.org/10.1007/BF02238484