The convergence behaviour of iterative methods on severely stretched grids

In this paper we examine the dramatic influence that a severe stretching of finite difference grids can have on the convergence behaviour of iterative methods. For the most important classes of iterative methods this phenomenon is considered for a simple model problem with various boundary conditions and an exponential grid. It is shown that grid compression near a Neumann boundary or in the centre can make the convergence of some methods extremely slow, whereas grid compression near a Dirichlet boundary can be very advantageous. More theoretical insight is obtained by analysing the spectrum of the Jacobi matrix for one- and two-dimensional problems. Several bounds on dominant eigenvalues of this matrix are given. The final conclusions are also applicable to problems with a variable diffusion coefficient and convection-diffusion equations solved by central difference schemes.