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A-posteriori error estimation and optimal adaptivity for fluid-structure interaction

Numerical simulations of fluid-structure interaction generally require vast computational resources. An interesting paradox in fluid-structure-interaction computations is that the computational cost is typically dominated by the subsystem that is of least practical interest, viz., the fluid. For realistic applications the fluid often consumes nearly all the computational resources, while practical engineering interest is restricted to the structural response. For instance, in [3] Farhat reports that for the investigation of the aeroelastic response of an F16 aircraft more than 98% of the total simulation time is spent inside the fluid solver and the mesh update algorithm. Goal-oriented a-posteriori error estimation and optimal adaptive refinement strategies provide an approach to bypass this paradox. Based on the solution of an appropriate dual problem, the contribution of local errors in the fluid solution to a particular functional (observation) of the structural displacement can be estabished. Only the regions in the fluid domain that have a pronounced influence on the error in the functional of interest need to be refined in the discretization. The general approach for goal-oriented a-posteriori error estimation has been developed by Becker and Rannacher [1] and Oden and Prudhomme [7,8]. The approach can be summarized as follows: Suppose that we are interested in the observation J(u) of the solution u ∈ U of the linear variational problem: a(u, v) = b(v) ∀v ∈ V, (1) with U and V certain Hilbert spaces, a : U × V → R a bounded bilinear form and b : V → R and J : U → R bounded functionals. Let Ũ ⊂ U and Ṽ ⊂ V represent conforming finite-element spaces. The corresponding finite-element approximation of J(u) is J(ũ) with ũ ∈ Ũ according to: a(ũ, ṽ) = b(ṽ) ∀ṽ ∈ Ṽ , (2)