Overlapping grids for the diffusion equation

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. © The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.