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

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

Van der Zee, K.G.
Van Brummelen, E.H.
De Borst, R.

Aerospace Engineering

Mechanics, Aerospace Structures & Materials

2010-03-31

In free-boundary problems, the accuracy of a goal quantity of interest depends on both the accuracy of the approximate solution and the accuracy of the domain approximation. We develop duality-based a posteriori error estimates for functional outputs of solutions of free-boundary problems that include both sources of error. The derivation of an appropriate dual problem (linearized adjoint) is, however, nonobvious for free-boundary problems. To derive an appropriate dual problem, we present the domain-map linearization approach. In this approach, the free-boundary problem is first transformed into an equivalent problem on a fixed reference domain after which the dual problem is obtained by linearization with respect to the domain map. We show for a Bernoulli-type free-boundary problem that this dual problem corresponds to a Poisson problem with a nonlocal Robin-type boundary condition. Furthermore, we present numerical experiments that demonstrate the effectivity of the dual-based error estimate and its usefulness in goal-oriented adaptive mesh refinement.

goal-oriented error estimation
a posteriori error estimation
Bernoulli free-boundary problem
domain-map linearization
linearized adjoint
adaptive mesh refinement

http://resolver.tudelft.nl/uuid:9a4ee49e-657d-4c4f-ad29-8b09c423f4ff

https://doi.org/10.1137/080741227

Society for Industrial and Applied Mathematics

1064-8275

SIAM Journal on Scientific Computing, 32 (2), 2010

Institutional Repository

journal article

(c) 2010 The Author(s)
Society for Industrial and Applied Mathematics