THE CONVERGENCE BEHAVIOR OF ITERATIVE METHODS ON SEVERELY STRETCHED GRIDS

In this paper we examine the dramatic influence that a severe stretchi ng of finite difference grids can have on the convergence behaviour of iterative methods. For the most important classes of iterative method s this phenomenon is considered for a simple model problem with variou s boundary conditions and an exponential grid. It is shown that grid c ompression near a Neumann boundary or in the centre can make the conve rgence 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- a nd two-dimensional problems. Several bounds on dominant eigenvalues of this matrix are given. The final conclusions are also applicable to p roblems with a variable diffusion coefficient and convection-diffusion equations solved by central difference schemes.