Error estimates for fractional semilinear optimal control on Lipschitz polytopes
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We adopt the integral definition of the fractional Laplace operator and analyze discretization techniques for a fractional, semilinear, and elliptic optimal control problem posed on a Lipschitz polytope. We consider two strategies of discretization: a semidiscrete scheme where control variables are not discretized – the so-called variational discretization approach – and a fully discrete scheme where control variables are discretized with piecewise constant functions. We discretize the corresponding state and adjoint equations with a finite element scheme based on continuous piecewise linear functions and derive error estimates. With these estimates at hand, we derive error bounds for the semidiscrete scheme on quasi-uniform and suitable graded meshes, and improve the ones that are available in the literature for the fully discrete scheme.
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