Multi-level hp-adaptivity: high-order mesh adaptivity without the difficulties of constraining hanging nodes

Abstract

The implementation of $$hp$$hp-adaptivity is challenging as hanging nodes, edges, and faces have to be constrained to ensure compatibility of the shape functions. For this reason, most $$hp$$hp-code frameworks restrict themselves to $$1$$1-irregular meshes to ease the implementational effort. This work alleviates these difficulties by introducing a new formulation for high-order mesh adaptivity that provides full local $$hp$$hp-refinement capabilities at a comparably small implementational effort. Its main idea is the extension of the $$hp$$hp-$$d$$d-method such that it allows for high-order overlay meshes yielding a hierarchical, multi-level $$hp$$hp-formulation of the Finite Element Method. This concept enables intuitive refinement and coarsening procedures, while linear independence and compatibility of the shape functions are guaranteed by construction. The proposed method is demonstrated to achieve exponential rates of convergence--both in terms of degrees of freedom and in run-time--for problems with non-smooth solutions. Furthermore, the scheme is used alongside the Finite Cell Method to simulate the heat flow around moving objects on a non-conforming background mesh and is combined with an energy-based refinement indicator for automatic $$hp$$hp-adaptivity.