Abstract: Optimal control problems,constrained by a state equation in the form of a partial differential equation (PDE),arise in many important applications, where one wants to steer the modelled process in order to have a solution close to some given target function.The discretized problems lead to a particular two-by-two block matrix form for which a very efficient preconditioner,leading to very tight eigenvalue bounds, will be presented. Various applications, such as in time-harmonic parabolic and Stokes equations and eddy current electromagnetic problems, will also be discussed.

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We study a model for lipid-bilayer membrane vesicles exhibiting phase separation, incorporating a phase field together with membrane fluidity and bending elasticity. We prove the existence of a plethora of equilibria in the large, corresponding to symmetry-breaking solutions of the Euler-Lagrange equations. We also numerically compute a special class of such solutions, namely those possessing icosahedral symmetry. We overcome several difficulties along the way. Due to inherent surface fluidity combined with finite curvature elasticity, neither the Eulerian (spatial) nor the Lagrangian (material) description of the model lends itself well to analysis. This is resolved via a singularity-free radial-map description, which effectively eliminates the grossly under-determined mid-plane deformation. We then use well known group-theoretic selection techniques combined with global bifurcation methods to obtain our results.

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