ABSTRACT
This chapter considers only the straight-forward extension of one-dimensional difference schemes to a rectangular mesh, where one has to interpolate for the points where a general boundary crosses the mesh. The accurate treatment of general two-dimensional problems will require the use of flexible meshes, constructed from quadrilaterals or triangles with mesh points lying on the boundary. In many practical problems, such as pollutant dispersal in a very irregular waterway, a rectangular mesh, with perhaps some local mesh refinement, is quite adequate for the accuracy that can reasonably be aimed at. For steady, linear problems in one or two dimensions, the systems of algebraic equations arising from finite difference or finite element methods will usually be solved by direct methods, or by 'black-box' iterative methods, including multigrid, especially as the pursuit of a maximum principle will have yielded a diagonally dominant matrix.
