Comparison of approximate shape gradients

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 BV...

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Bibliographic Details
Published inBIT Numerical Mathematics Vol. 55; no. 2; pp. 459 - 485
Main Authors Hiptmair, R., Paganini, A., Sargheini, S.
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.06.2015
Subjects
Computational Mathematics and Numerical Analysis
Mathematics
Mathematics and Statistics
Numeric Computing
Duality techniques
Shape calculus
Finite element approximations
49Q12
65D15
Shape gradients
65N30
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Summary: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.
ISSN:0006-3835
1572-9125
DOI:10.1007/s10543-014-0515-z