We review techniques of numerical bifurcation and stability analysis with examples from computational fluid dynamics and biology. The methodology allows insight into the complete dynamics of nonlinear PDE systems, where standard simulation tool chains leave the question of existence, proximity and stability of multiple solutions open. The main bottleneck in the method are large and sparse linear systems of equations and eigenvalue problems
arising from the discretized steady-state PDE. The use of HPC is therefore attractive to increase the achievable resolution, but remains challenging because nonsymmetric and indefinite systems need to be solved. The `hybrid multi-level solver' HYMLS is a robust multi-level incomplete factorization technique that was designed for this particular class of problems. HYMLS has an intuitive geometric interpretation and good parallelization properties. We present some performance results of a prototypical implementation based on MPI and the Trilinos software. The eigenvalue problems that arise are solved using the Jacobi-Davidson method as implemented in the SPPEXA ESSEX~\cite{essex} project's phist library (https://www.bitbucket.org/essex/phist)
