What are C and h?: inequalities for the analysis and design of finite element methods
Computer Methods in Applied Mechanics and Engineering
Shapes and geometries: analysis, differential calculus, and optimization
Shapes and geometries: analysis, differential calculus, and optimization
Efficient treatment of stationary free boundary problems
Applied Numerical Mathematics - Selected papers from the first Chilean workshop on numerical analysis of partial differential equations (WONAPDE 2004)
On Convergence in Elliptic Shape Optimization
SIAM Journal on Control and Optimization
SIAM Journal on Scientific Computing
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We develop duality-based a posteriori error estimates for functional outputs of solutions of free-boundary problems via shape-linearization principles. To derive an appropriate dual (linearized adjoint) problem, we linearize the domain dependence of the very weak form and goal functional of interest using techniques from shape calculus. We show for a Bernoulli-type free-boundary problem that the dual problem corresponds to a Poisson problem with a Robin-type boundary condition involving the curvature. Moreover, we derive a generalization of the dual problem for nonsmooth free boundaries which includes a natural extension of the curvature term. The effectivity of the dual-based error estimate and its usefulness in goal-oriented adaptive mesh refinement is demonstrated by numerical experiments.