| Issue |
ESAIM: M2AN
Volume 59, Number 4, July-August 2025
|
|
|---|---|---|
| Page(s) | 2141 - 2170 | |
| DOI | https://doi.org/10.1051/m2an/2025058 | |
| Published online | 23 July 2025 | |
Convergence analysis of approximate shape gradients for shape optimization in parabolic problems
1
School of Mathematical Sciences, Shanghai Jiao Tong University, 200240 Shanghai, P.R. China
2
School of Mathematical Sciences, East China Normal University, 200241 Shanghai, P.R. China
3
Key Laboratory of MEA (Ministry of Education) & Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice & School of Mathematical Sciences, East China Normal University, 200241 Shanghai, P.R. China
* Corresponding author: sfzhu@math.ecnu.edu.cn
Received:
25
November
2024
Accepted:
23
June
2025
Numerical approximations of shape gradients and their applications have recently caused much interest in the shape optimization community. In this paper, existing research results on shape optimization governed by steady problems (e.g., elliptic problems in Hiptmair et al. [BIT Numer. Math. 55 (2015) 459–485]) are extended to those governed by parabolic problems. Convergence analysis is presented for numerical approximations of shape gradients associated with a parabolic problem. Both the backward Euler scheme and the backward differentiation formula are employed for time discretization, and the Galerkin finite element method is used for spatial discretization of the parabolic state and adjoint problems. The error of the distributed shape gradient is shown to have a higher convergence order in the mesh-size than that of the boundary type. A priori error estimates with respect to the time step-size are also presented. Numerical examples are provided to illustrate the theoretical results.
Mathematics Subject Classification: 49M25 / 49M41 / 65M12 / 65M60
Key words: Shape optimization / parabolic problem / shape gradients / finite element method / error estimate
© The authors. Published by EDP Sciences, SMAI 2025
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