AbstractThe structure of the state space of distributed, nonlinear systems, i.e. the solutionspace of parameter dependent partial differential and integral equations, can be analyzedby recently emerged, powerful techniques of computer-aided functional analysis. Pre-eminent is discretization by the method of sub domains, finite element basis functions, and Galerk in's method of weighted residuals: for steady states the result is large sets of nonlinear algebraic equations. Free boundaries and irregular domains are handled efficientlyby appropriate spatial parametrization and flexible isoparametrie mapping into a computational domain. The algebraic equation set is solved by Newton's method or an efficient variant; because the matrices from partial differential equations are large,sparse and structured, vectorized frontal solvers are cost-effective on advanced computers. Families of solutions are generated by continuation in parameters, with special adaptive techniques toget around singular states(bifurcation and turning points). Stability of steady states is examined by analyzing response to normal modes of small disturbances, i.e. eigenanalysis from linear stability theory. The account here is by two examples drawn from electrohydro statics and magnetohydros tatics.