Error Estimations and Adaptive Techniques for Nonlinearized Parametrized Equations.
PITTSBURGH UNIV PA INST FOR COMPUTATIONAL MATHEMATICS AND APPLICATIONS
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Many problems in science and engineering concern the determination of steady-state equilibrium solutions of nonlinear equations. In general, for such problems, interest centers not so much of the computation of a few specific equilibria rather than on an assessment of the response of the system to the action of various external or internal influence quantities. In other words, we are interested in the effect of changes of the values of certain parameters upon the computed equilibria. Thus, other than in the typical linear case, for nonlinear problems we usually have to consider equations of the form which depend nonlinearly not only on the state variable z but also on a parameter vector lambda. Typically, z varies in some infinite-dimensional space Z while lambda belongs to a space lambda with some finite dimension m or z of 1.1 for a few a priori specified parameter vectors lambda. Instead, we have to look at the solutions of 1.1 as points z, lambda in the product X Z x lambda of the state and and parameter space. Under fairly general conditions, the set of all solutions z, lambda of 1.1 in X forms a smooth surface -- or more precisely an m-dimensional differentiable manifold -- in that space. When 1.1 represents the equilibrium equation of a mechanical system, this manifold has been called the equilibrium surface of the system.
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