On the Discretization Error of Parametrized Nonlinear Equations

Many applications lead to nonlinear, parameter dependent equations H(y,t) = y₀, where H:Y × T → Y, y₀ ∈ rgeH, and the state space Y is infinite-dimensional while the parameter space T has finite dimension. The case T = 1 is of special interest in connection with continuation methods. For this case, a general theory is developed which provides for the existence of solution paths of a rather general class of such equations and of their finite-dimensional approximations, and which allows for an assessment of the error between these paths. A principal tool in this analysis is the theory of nonlinear Fredholm operators. The results cover a more general class of operators than the mildly nonlinear mappings to which other approaches appear to be restricted.