25.01.2011_03 - C omputing 61, 285-305 (1998) ~ 1 ~...

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Computing 61, 285-305 (1998) ~1~ @ 8pringer-Veflag 1998 Printed in Austria A Finite Volume Scheme for Solving Elliptic Boundary Value Problems on Composite Grids M. J. H. Anthonissen, B. van 't Hof, Eindhoven, and A. A. Reusken, Aachen Received March 5, 1998; revised July 22, 1998 Abstract We present a finite volume scheme for solving elliptic boundary value problems with solutions that have one or a few small regions with high activity. The scheme results from combining the local defect correction method (LDC), introduced in [11], with standard finite volume discretizations on a global coarse and on local fine uniform grids. The iterative discretization method that is obtained in this way yields a discrete approximation of the continuous solution on a composite grid. For the LDC method in its standard form, the discrete conservation property, which is one of the main attractive features of a finite volume method, is lost for the composite grid approximation. For the modified LDC method we present here, discrete conservation holds for the composite grid solution, too. AMS Subject Classifications: 65N22, 65N50. Key words: Elliptic problems, finite volume methods, local defect correction, composite grids. I. Introduction Many boundary value problems produce solutions that have highly localized properties. In this paper we consider elliptic boundary value problems with solutions that have one or a few small regions with high activity. We study a finite volume discretization method based on a combination of standard finite volume discretizations on several uniform grids with different grid sizes that cover different parts of the domain. At least one grid should cover the entire domain; the meshsize of this global coarse grid is chosen in agreement with the relatively smooth behavior of the solution outside the high activity regions. Apart from this global coarse grid, one or several local grids are used which are also uniform. Each of these local grids covers only a (small) part of the domain and contains a high activity region. The meshsizes of the local grids are chosen in agreement with the behavior of the solution in the corresponding high activity region. In this way, every part of the domain can be covered by a
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286 M.J.H. Anthonissen et al. (locally) uniform grid with a meshsize that is in agreement with the behavior of the continuous solution in that part of the domain. This refinement strategy is known as locally uniform grid refinement. The solution is approximated on the composite grid, which is the union of the uniform subgrids. Note that such composite grids are highly structured and hence very simple data structures can be used. In [11], Hackbusch introduced the local defect correction method (LDC) for approximating the continuous solution of elliptic boundary value problems on a composite grid. In this method, which is an iterative process, a basic global discretization is improved by local discretizations defined in the subdomains. This update of the coarse grid solution is achieved by putting a defect correction term in the right hand side of the coarse grid problem. At each iteration step,
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25.01.2011_03 - C omputing 61, 285-305 (1998) ~ 1 ~...

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