IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v42y1996i4p595-605.html
   My bibliography  Save this article

A proof of convergence for the combination technique for the Laplace equation using tools of symbolic computation

Author

Listed:
  • Bungartz, H.
  • Griebel, M.
  • Röschke, D.
  • Zenger, C.

Abstract

For a simple model problem—the Laplace equation on the unit square with a Dirichlet boundary function vanishing for x = 0, x = 1 and y = 1, and equaling some suitable g(x) for y = 0—we present a proof of convergence for the so-called combination technique, a modern, efficient and easily parallelizable sparse grid solver for elliptic partial differential equations that recently gained importance in fields of applications like computational fluid dynamics. For full square grids with meshwidth h and O(h−2) grid points, the order O(h2) of the discretization error was shown in (Hofman, 1967), if g(x) ϵC2[0, 1]. In this paper, we show that the error of the solution produced by the combination technique on a sparse grid with only O((h−1log2(h−1)) grid points is of the order O(h2log2(h−1)), if gϵC4[0, 1], and g(0) = g(1) = g″(0) = g″(1) = 0. The crucial task of the proof, i.e. the determination of the discretization error on rectangular grids with arbitrary meshwidths in each coordinate direction, is supported by an extensive and interactive use of Maple.

Suggested Citation

  • Bungartz, H. & Griebel, M. & Röschke, D. & Zenger, C., 1996. "A proof of convergence for the combination technique for the Laplace equation using tools of symbolic computation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 42(4), pages 595-605.
    • Handle: RePEc:eee:matcom:v:42:y:1996:i:4:p:595-605
    DOI: 10.1016/S0378-4754(96)00036-5
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475496000365
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/S0378-4754(96)00036-5?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:matcom:v:42:y:1996:i:4:p:595-605. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.