Abstract
In the present paper, numerical solving of the double-periodic elliptic eigenvalue problems
is considered regarding special symmetry properties. At first, subspacesV with the desired symmetry are constructed then a classical Ritz method is applied for the discretization inV and the resulting finite-dimensional bifurcation problem is solved by an algorithm proposed by Keller and Langford representing anumerical implementation of the Ljapunov-Schmidt procedure. Iff(u) is an entire function or a polynomial andV is an algebra then the computed solutions reveal to be stable with respect to perturbations of less symmetry. Some examples demonstrate the efficiency of the procedure.
Zusammenfassung
In dieser Arbeit wird die numerische Lösung doppelt-periodischer elliptischer Eigenwertprobleme
untersucht unter besondere Berücksichtigung von Symmetrieeigenschaften. Zuerst werden UnterräumeV mit der gewünschten Symmetrie konstruiert und dann ein klassisches Ritz-Verfahren zur Diskretisierung inV verwendet. Das resultierende endlichdimensionale Gleichungssystem wird mit einem Algorithmus von Keller und Langford gelöst, der eine numerische Implementierung des Verfahrens von Ljapunov und Schmidt darstellt. Wennf(u) eine ganze Funktion oder ein Polynom ist undV eine Algebra, dann ist die Ritz-Approximation stabil gegenüber Störungen von geringerer Symmetrie. Einige Beispiele demonstrieren die Wirkungsweise des Verfahrens.
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Gekeler, E.W. On the numerical solution of double-periodic elliptic eigenvalue problems. Computing 43, 97–114 (1989). https://doi.org/10.1007/BF02241855
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DOI: https://doi.org/10.1007/BF02241855