Conjugate gradient type methods for semilinear elliptic problems with symmetry

We study block conjugate gradient methods in the context of continuation methods for bifurcation problems. By exploiting symmetry in certain semilinear elliptic differential equations, we can decompose the problems into small ones and reduce computational cost. On the other hand, the associated centered 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 properties allow the discrete pure mode solution paths branching from a multiple bifurcation point [0, λm,n] of the centered difference analogue of the original problem to be represented by the solution path branching from the first simple bifurcation point (0, μ1,1) of the counterpart of the reduced problem. Thus, the structure of a multiple bifurcation is preserved in discretization, while its treatment is reduced to those for simple bifurcation of problems on subdomains. In particular, we can adapt the continuation-Lanczos algorithm proposed in [1] to trace simple solution paths. Sample numerical results are reported.