On Hybrid Methods for Bifurcation and Center Manifolds for General Operators

This presentation uses few basic concepts of numerical functional analysis and approximation theory as the main tools to prove convergence and stability for stationary problems. It applies to a general class of operator equations and general discretization methods. This allows an extension to numerical bifurcation studies, including Hopf bifurcation and center manifold results, for finite difference-, finite element- and spectral methods for general operators. In particular, partial differential equations (PDEs) as reaction-diffusion-systems and Navier-Stokes equations are included. The basic idea is to present an approach as simple as possible but as complex as necessary to cover all these types of problems and their discretizations with reasonably basic concepts. For the first time, the full cycle of qualitative and quantitative results, starting from PDEs via convergent discretization and post-processing back to the bifurcation scenarios in the original equation, is presented. A Г-equi-variant example in biological pattern formation is included. Finally a C ++ — program system with similarly general goals is indicated.