Abstract
An abstract framework ofauxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxiliary space (that may be roughly understood as a nonnested coarser space). An optimal multigrid preconditioner is then obtained for a discretized partial differential operator defined on an unstructured grid by using an auxiliary space defined on a more structured grid in which a furthernested multigrid method can be naturally applied. This new technique makes it possible to apply multigrid methods to general unstructured grids without too much more programming effort than traditional solution methods. Some simple examples are also given to illustrate the abstract theory and for instance the Morley finite element space is used as an auxiliary space to construct a preconditioner for Argyris element for biharmonic equations. Some numerical results are also given to demonstrate the efficiency of using structured grid for auxiliary space to precondition unstructured grids.
Zusammenfassung
Das abstrakte Konzept derHilfsraummethode wird vorgeschlagen, und als Anwendung eine optimale Mehrgittertechnik für allgemeine unstrukturierte Gitter entwickelt. Die Hilfsraummethode ist eine (nichtgeschachtelte) Zweigitter-Präkonditionierungstechnik, bestehend aus einem einfachen Glättungsverfahren und einem Hilfsgitter, das etwa einem nichtgeschachtelten Grobgitter entspricht. Zu einer partiellen Differentialgleichung, die auf einem unstrukturierten Gitter diskretisiert ist, wird eine optimale Mehrgittermethode konstruiert, die einen Hilfsraum mit einem besser strukturierten Gitter verwendet. In diesem ist ein Mehrgitterverfahren mit geschachtelten Räumen anwendbar. Die neue Technik ermöglicht es, Mehrgittermethoden für unstrukturierte Gitter anzuwenden, die nur wenig mehr Programmieraufwand erfordern als traditionelle Methoden. Einige einfache Beispiele sollen die abstrakte Theorie illustrieren. Als Beispiel wird der Morley-FE-Raum als Hilfsraum zur Präkonditionierung der Argyris-Elemente für die biharmonische Gleichung benutzt. Einige numerische Resultate demonstrieren die Effizienz des Einsatzes des strukturierten Hilfsraumes.
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This work was partially supported by National Science Foundation, DMS94-03915-1.
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Xu, J. The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing 56, 215–235 (1996). https://doi.org/10.1007/BF02238513
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DOI: https://doi.org/10.1007/BF02238513