Adaptive Pseudo–transient Continuation for Nonlinear Steady State Problems

Pseudo–transient continuation methods are quite popular for the numerical solution of steady state problems, typically in PDEs. They are based on an embedding into a time dependent initial value problem. In the presence of dynamical invariants the Jacobian matrix of the nonlinear equation system is bound to be singular. The paper presents a convergence analysis which takes this property into account – in contrast to known approaches. On the basis of the new analysis adaptive algorithms are suggested in detail. These include a variant with Jacobian approximations as well as inexact pseudo–transient continuation, both of which play an important role in discretized PDEs. Numerical experiments are left to future work.