LARGE-SCALE CONTINUATION AND NUMERICAL BIFURCATION FOR PARTIAL-DIFFERENTIAL EQUATIONS

In this paper the problem of computing bifurcation diagrams for large- scale nonlinear parameter-dependent steady state systems which arise f ollowing the spatial discretization of semilinear PDEs is investigated . A continuation algorithm which employs a preconditioned version of t he recursive projection method (RPM) is presented. The RPM is often ex pensive when it is used in conjunction with the numerical method of li nes. Preconditioning the Jacobian of the underlying fixed point operat or results in an algorithm (the preconditioned recursive projection me thod (PRPM)) which is capable of efficiently computing equilibrium sol ution diagrams of large stiff systems. For many PDE problems the PRPM is a fast and effective means of detecting both steady state and Hopf bifurcation along a branch of solutions. A description of the performa nce of the PRPM when applied to two numerical examples is given.