Background : Nonlinear differential equations (NDEs) govern a wide class of phenomena in fluid mechanics, structural engineering, and mathematical biology. Classical analytical methods are seldom applicable to these problems, necessitating robust numerical frameworks.
Objective : This study presents a systematic evaluation of advanced finite element methods (FEM) for solving nonlinear differential equations, with emphasis on adaptive mesh refinement, iterative solvers, and convergence behaviour.
Methods : Four FEM formulations are compared: standard Galerkin, hp-adaptive, discontinuous Galerkin (DG), and isogeometric analysis (IGA). Benchmark problems including Burgers' equation and the incompressible Navier–Stokes equations are solved and analysed.
Results : The hp-FEM and IGA approaches achieved convergence rates exceeding 3.0 on smooth problems, with L² errors below 5 × 10⁻⁶ on meshes of moderate density. Computational costs scaled favourably with the proposed preconditioning strategies.
Conclusion : Advanced FEM formulations offer superior accuracy and efficiency for nonlinear problems. Future work should focus on GPU-accelerated solvers and data-driven mesh adaptation strategies.