The numerical solution of a control problem governed by a phase filed model

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Abstract
The application of a multilevel Newton method for the numerical solution of an optimal control problem governed by a phase field model is investigated. The phase field model is used to describe the phase change (e.g. liquid to solid) of materials and is given as a system of two semilinear parabolic equations. For the solution of the optimal control problem the necessary optimality conditions are reformulated as a compact fixed point problem. In this formulation, controls, states, and adjoints are viewed as independent variables, the state equation is treated as a constraint. A multilevel Newton method due to Atkinson and Brakhage is applied for its solution. The multilevel Newton method operates on a sequence of grids and uses inexpensive coarse grid information to obtain good approximations for the Newton step. If the coarse grid is sufficiently fine, this yields a fast linearly convergent method which considerably reduces the amount of work per iteration and also requires fewer data to handle. Other advantages of this approach are that the reformulation as a compact fixed point problem yields a decoupling of the phase field model and generates functions that only involve the solution of linear parabolic equations. Numerical examples indicate that the work per iteration of the resulting algorithm is linear in the number of variables, if fast methods for the solution of the linear PDEs are used