Goal-Oriented Error Estimation and Adaptivity for Free-Boundary Problems: The Shape-Linearization Approach

Abstract. 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 dualbased error estimate and its usefulness in goal-oriented adaptive mesh refinement is demonstrated by numerical experiments. Key words. goal-oriented error estimation, a posteriori error estimation, Bernoulli freeboundary problem, shape derivative, shape differential calculus, linearized adjoint, adaptive me...

• Type: Journal
• Year: 2010
• Where: SIAMSC
• Authors: K. G. van der Zee, E. H. van Brummelen, R. de Borst
• Keywords: Adaptive Mesh Refinement, Dual Problem, Posteriori Error, SIAMSC 2010