  Removing geometrical details from a complex domain is a classical operation
in computer aided design. This procedure simplifies the meshing process, and it
enables faster simulations with less memory requirements. However, depending on
the partial differential equation that one wants to solve, removing some
important geometrical features may greatly impact the solution accuracy.
Unfortunately, the effect of geometrical simplification on the accuracy of the
problem solution is often neglected or its evaluation is based on engineering
expertise, only due to the lack of reliable tools. It is therefore important to
have a better understanding of the effect of geometrical model simplification,
also called defeaturing, to improve our control on the simulation accuracy
along the design and analysis phases. In this work, we consider as a model
problem the Poisson equation on a geometry with Neumann features, we consider
some finite element discretization of it, and we build an adaptive strategy
that is twofold. Firstly, it is able to perform geometrical refinements, that
is, to choose at each iteration step which geometrical feature is important to
obtain an accurate solution. Secondly, it performs standard mesh refinements;
since the geometry changes at each iteration, the algorithm is designed to be
used with an immersed method. To drive this adaptive strategy, we introduce an
a posteriori estimator of the energy error between the exact solution defined
in the exact fully-featured geometry, and the numerical approximation of the
solution defined in the defeatured geometry. The reliability of the estimator
is proven for very general (potentially trimmed multipatch) geometric
configurations, and in particular for IGA with hierarchical B-splines. Finally,
numerical experiments are performed to validate the presented theory and to
illustrate the capabilities of the proposed adaptive strategy.
