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

TL;DR

This paper introduces an $h$-adaptive finite element method using Kelly error estimation for solving the Falkner-Skan equation, effectively capturing boundary layer behavior and accurately computing skin friction across various flow conditions.

Contribution

The paper develops and applies an $h$-adaptive finite element approach with Kelly error estimator for efficient Falkner-Skan equation solutions, improving accuracy in boundary layer analysis.

Findings

Accurate resolution of boundary layer behavior.

Effective computation of skin friction coefficient.

Robustness across diverse flow parameters.

Abstract

This paper details the development and application of an $h$-adaptive finite element method for the numerical solution of the \textit{Falkner-Skan equation}. A posteriori error estimation governs the adaptivity of the mesh, specifically the well-established \textit{Kelly error estimator}, which utilizes the jump in the gradient across elements. The implementation of this method allowed for accurate and efficient resolution of the boundary layer behavior characteristic of Falkner-Skan flows. Numerical solutions were obtained across various wedge flow parameters, encompassing favorable and adverse pressure gradients. A key focus of this study was the precise computation of the skin friction coefficient, a critical parameter in boundary layer analysis, across this diverse range of flow conditions. The results are presented and discussed, demonstrating the robustness and accuracy of the adaptive finite element approach for this class of nonlinear boundary layer problems.