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In the first part, we present continuous higher order finite element discretizations for the model problem and we derive error estimates. In the second part, we discuss Newton iterative methods based on residual-based line-search and error-oriented globalization, which are employed for the numerical solution of the produced nonlinear algebraic system. Third, we formulate the original problem as a saddle point problem in the frame of augmented Lagrangian techniques and present two iterative methods for its solution. We conduct a systematic investigation of all solution algorithms. These algorithms are compared with respect to computational cost and their efficiency. 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