Discretizing the diffusion equation on unstructured polygonal meshes in two dimensions

We derive a discretization of the two-dimensional diffusion equation for us e with unstructured meshes of polygons. The scheme is presented in r-z geom etry, but can easily be applied to x-y geometry. The method is "node" - or "point" -based and is constructed using a finite volume approach. The schem e is designed to have several important properties, including second-order accuracy. convergence to the exact result as the mesh is refined (regardles s of the smoothness of the grid), and preservation of the homogeneous linea r solution. Its principle disadvantage is that, in general, it generates an asymmetric coefficient matrix, and therefore requires more storage and the use of non-traditional, and sometimes slowly-converging. iterative linear solvers. On an unstructured triangular grid in x y geometry. the scheme is equivalent to the linear continuous finite element method with "mass-matrix lumping". We give computational examples that demonstrate the accuracy and convergence properties of the new scheme relative to other schemes. (C) 20 01 Elsevier Science Ltd. All rights reserved.