One of the challenges faced by hp-adaptive methods is their implementational complexity and/or computational cost. Particularly, there are two bottlenecks identified : the data structures to handle hp-refined meshes are often complex, and the challenging design of robust and automatic strategies.
Our approach is based on a hierarchical data structure developed recently by N. Zander et al. It allows arbitrarily irregular meshes by eliminating the difficulties of hanging nodes via Dirichlet patches.
The main contribution is the design of a novel automatic and robust hp-strategy. We first design the strategy for elliptic problems and show its efficiency on benchmark tests (1D, 2D, and 3D). We then show how our strategy can be (easily) extended to non-elliptic problems, in particular for Helmholtz and convection-dominated diffusion-convection problems. Furthermore, in many engineering applications, the aim is only a feature of the solution (instead of the whole solution). Therefore, specific meshes can be generated to fit that purpose. We show how we adapt the strategy for goal-oriented applications and present the results for various 2D examples.
The main advantages and limitations of our approach will be discussed.
Romain Duboscq, Ariane Trescases