For a bounded linear operator on a Banach space, we study approximation of the spectrum and pseudospectra in the Hausdorff distance. We give sufficient and necessary conditions in terms of pointwise convergence of appropriate spectral quantities.
A question, which arises frequently in shape optimal design, is the convergence of domains. If the objective function is defined by using the solution of a PDE with boundary conditions, then also the convergence of the boundary is of importance. In this paper a criterion for a set of domains is defined, such that from [Omega_n] --> Omega follows [Gamma_n] --> Gamma if one is restricting to this set of domains. Moreover it is proved that this criterion is sharp, meaning that if [Omega_n] --> Omega implies [Gamma_n] --> Gamma holds for any sequence of this set, then this criterion has to be fulfilled. A similar criterion for the convergence of the Lebesgue measure of the boundaries my(Gamma_n) --> my(Gamma) is given.
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