Eff. Botta et Fw. Wubs, THE CONVERGENCE BEHAVIOR OF ITERATIVE METHODS ON SEVERELY STRETCHED GRIDS, International journal for numerical methods in engineering, 36(19), 1993, pp. 3333-3350
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.