Automated h -adaptivity for finite element approximations of the Falkner–Skan equation

Abstract This paper details the development and application of an h -adaptive finite element method for the numerical solution of the Falkner–Skan equation . While traditional techniques, such as the shooting method, often rely on arbitrary trial parameters to enforce boundary conditions, the novelty of this approach lies in its direct incorporation of physically meaningful boundary conditions without any ad-hoc intermediary assumptions. A posteriori error estimation governs the adaptivity of the mesh, specifically utilizing the well-established Kelly error estimator to evaluate the jump in the solution gradient across element interfaces. Numerical solutions were successfully obtained across a wide range of wedge flow parameters, encompassing both favorable and adverse pressure gradients. A significant result of this study is the high-precision computation of the skin friction coefficient-a critical parameter in boundary layer analysis-which was accurately captured with targeted mesh refinement in regions of steep gradients. The findings conclude that this automated local mesh adaptivity strategy provides an easy-to-implement and reliable method over standard uniform meshing. Ultimately, this work demonstrates that the adaptive finite element framework is a highly reliable and extensible tool for resolving complex, localized behaviors in nonlinear fluid dynamics.