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IMA Journal of Numerical Analysis Advance Access originally published online on August 28, 2006
IMA Journal of Numerical Analysis 2007 27(3):550-575; doi:10.1093/imanum/drl022
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© The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Overlapping grids for the diffusion equation

Dugald B. Duncan{dagger}

Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, UK

Yiqi Qiu{ddagger}

Edinburgh Petroleum Services Ltd, Research Park, Riccarton, Edinburgh EH14 4AP, UK

{dagger} Email: d.b.duncan{at}ma.hw.ac.uk

{ddagger} Email: Y.Qiu{at}ma.hw.ac.uk

Received on 23 July 2004. Revised on 14 June 2006.


  Abstract

We examine the use of nonmatching, overlapping grids for the approximate solution of time-dependent diffusion problems with Neumann boundary conditions. This problem arises as a model of the so-called well test analysis of oil and gas reservoirs, which has geometry modelling requirements that make overlapping grids particularly suitable. We describe the problem and the overlapping grid approximation, and then carry out a stability and convergence analysis in one space dimension (1D). We show that for suitable schemes, stability is relatively easy to establish in much more general situations. Convergence is less easy to generalise, but we demonstrate that 2D approximations appear to have the same convergence behaviour as their 1D counterparts.

Key Words: overlapping grids; domain decomposition; well test analysis


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