Conjugate gradient type methods for semilinear elliptic problems with symmetry

We study block conjugate gradient methods in the context of continuation me thods for bifurcation problems. By exploiting symmetry in certain semilinea r elliptic differential equations, we can decompose the problems into small ones and reduce computational cost. On the other hand, the associated cent ered difference discretization matrices on the subdomains are nonsymmetric. We symmetrize them by using simple similarity transformations and discuss some basic properties concerning the discretization matrices. These propert ies allow the discrete pure mode solution paths branching from a multiple b ifurcation point [0, lambda(m,n)] of the centered difference analogue of th e original problem to be represented by the solution path branching from th e first simple bifurcation point (0, mu(1,1)) of the counterpart of the red uced problem. Thus, the structure of a multiple bifurcation is preserved in discretization, while its treatment is reduced to those for simple bifurca tion of problems on subdomains. In particular, we can adapt the continuatio n-lanczos algorithm proposed in [1] to trace simple solution paths. Sample numerical results are reported. (C) 1999 Elsevier Science Ltd. All rights r eserved.