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Discretization of elliptic differential equations on curvilinear bounded domains with sparse grids

Diskretisierung elliptischer Differentialgleichungen auf krummlinig berandeten Gebieten mit dünnen Gittern

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Abstract

Elliptic differential equations can be discretized with bilinear finite elements. Using sparse grids instead of full grids, the dimension of the finite element space for the 2D problem reduces fromO(N 2) toO (N logN) while the approximation properties are nearly the same for smooth functions. A method is presented which discretizes elliptic differential equations on curvilinear bounded domains with adaptive sparse grids. The grid is generated by a transformation of the domain. This method has the same behaviour of convergence like the sparse grid discretization on the unit square.

Zusammenfassung

Elliptische Differentialgleichungen werden häufig mittels bilinearer Finite-Elemente diskretisiert. Verwendet man hierfür dünne Gitter anstelle voller Gitter, so reduziert sich die Dimension des Finite-Element-Raumes eines 2D-Problems vonO(N 2) aufO(N logN). Die Approximationseigenschaften bleiben jedoch nahezu erhalten. Es wird eine Methode zur Diskretisierung elliptischer Differentialgleichungen auf krummlinig berandeten Gebieten mittels adaptiver dünner Gitter vorgestellt. Das Gitter wird durch eine Transformation des Gebietes erzeugt. Das Konvergenzverhalten dieses Verfahrens kommt dem der Dünngitter-Diskretisierung auf dem Einheitsquadrat gleich.

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  1. Inst. für Informatik, Technische Universität München, D-80290, München, Germany

    T. Dornseifer & C. Pflaum

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  1. T. Dornseifer
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  2. C. Pflaum
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Dornseifer, T., Pflaum, C. Discretization of elliptic differential equations on curvilinear bounded domains with sparse grids. Computing 56, 197–213 (1996). https://doi.org/10.1007/BF02238512

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  • DOI: https://doi.org/10.1007/BF02238512

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