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Comparison of Approximate Shape Gradients

by R. Hiptmair and A. Paganini and S. Sargheini

(Report number 2013-30)

Abstract
Shape gradients of PDE constrained shape functionals can be stated in two equivalent ways. Both rely on the solutions of two boundary value problems (BVPs), but one involves integrating their traces on the boundary of the domain, while the other evaluates integrals in the volume. Usually, the two BVPs can only be solved approximately, for instance, by finite element methods. However, when used with finite element solutions, the equivalence of the two formulas breaks down. By means of a comprehensive convergence analysis, we establish that the volume based expression for the shape gradient generally offers better accuracy in a finite element setting. The results are confirmed by several numerical experiments.

Keywords: Shape Gradients, Shape Calculus, Finite Element Approximations

@Techreport{HPS13_527,
  author = {R. Hiptmair and A. Paganini and S. Sargheini},
  title = {Comparison of Approximate Shape Gradients},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-30},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-30.pdf },
  year = {2013}
}

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