Abstract
A method for solving the elliptic second order mixed boundary value problem is discussed. The finite element problem is divided into subproblems, associated with subregions into which the region has been partitioned, and an auxiliary problem connected with intersect curves. The subproblems are solved directly, while the auxiliary problem is handled by a conjugate gradient method. The rate of the convergence of the cg-method is discussed also for the cases when the Neumann and Dirichlet boundary conditions change at points belonging to the intersecting curves. Results from numerical experiments are also reported.
Zusammenfassung
Es wird eine Methode zur Lösung elliptischer Probleme zweiter Ordnung mit gemischten Randbedingungen diskutiert. Das Finite-Elemente-Problem wird in Teilprobleme aufgespalten. Daraus resultieren Unterprobleme, welche auf die durch die Aufteilung entstehenden Teilgebiete bezogen sind, sowie ein Hilfsproblem, das mit den Trennkurven zusammenhängt. Die Unterprobleme werden direkt gelöst, das Hilfsproblem wird mittels einer Konjuguerten-Gradienten-Methode behandelt. Die Konvergenzgeschwindigkeit der KG-Methode wird auch für den Fall analysiert, daß Neumann und Dirichlet Randbedingungen auf Punkten der Trennkurven wechseln. Numerische Erfahrungen liegen vor.
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Mróz, M. Domain decomposition method for elliptic mixed boundary value problems. Computing 42, 45–59 (1989). https://doi.org/10.1007/BF02243142
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DOI: https://doi.org/10.1007/BF02243142