Goal-oriented Error Estimation for Free-boundary Problems Using the Exact Shape-linearized Adjoint

Since the late 1990s, goal-oriented error estimation and goal-oriented adaptive methods have been developed to control the discretization error in goal functionals of the solution1,2. These methods have mostly been applied to linear and nonlinear problems in solid and fluid mechanics. An important recent development is the extension of goal-oriented adaptive methods to multiphysics problems involving multiple coupled boundary value problems3,4,5. Some authors have also considered an extension to nonlinear free-boundary problems such as fluid–structureinteraction6,7,8. This has shown to be quite difficult owing to the free-boundary character. Free-boundary problems elude the goal-oriented-error estimation framework because such problems are posed on an a priori unknown domain. This means that the standard variational form of free-boundary problems contains an unusual domain-dependent nonlinearity. The derivation of the linearized-adjoint (dual) operator is therefore highly nontrivial. Indeed, the approaches in Refs. 6 and 7 try to bypass the derivation of the linearized adjoint by neglecting the domain-dependence or approximating it using a finite difference approach. We note that in Ref. 8 an exact linearized adjoint is derived by transforming the problem to a fixed reference domain. Unfortunately, the corresponding dual problem contains nonstandard and nonlocal interior and boundary terms, which are inconvenient from an implementation