Chapter Three - Anisotropic recovery-based error estimators and mesh adaptation for real-life engineering innovation

This chapter presents an overview of anisotropic mesh adaptation techniques driven by recovery-based a posteriori error estimators. The first part outlines the theoretical foundation for anisotropic error estimation and the construction of metric-based adapted meshes in a steady context. The methodology is then extended to time-dependent problems by coupling mesh adaptation with adaptive time stepping, in a unified space-time framework. The approach is tested on three representative engineering applications, namely structural topology optimization, microstructured material design, and unsteady fluid dynamics, demonstrating the effectiveness in capturing directional features in space and heterogeneities in time. The proposed strategy offers practical advantages in terms of computational efficiency, broad applicability, and ease of integration into existing numerical solvers.