Spectral element method for the nonlinear Fokker–Planck equations in the whole line

For nonlinear partial differential equations on the whole line, where the solution exhibits Gaussian-like characteristics and severe oscillations, we propose a spectral element method based on non-uniform meshes. To approximate the highly oscillatory part of the solution on bounded subdomains, we employ Legendre spectral elements, while for the part of the solution that decays at infinity on unbounded subdomains, we utilize Laguerre functions. A novel composite spectral element scheme is developed, featuring movable common boundaries between adjacent subdomains. This design makes the scheme particularly effective for Gaussian-type solutions governed by nonlinear partial differential equations. The convergence and stability of the proposed algorithm are established through a newly constructed composite quasi-orthogonal approximation framework over the entire real line. Numerical results confirm the effectiveness of the algorithm and are in excellent agreement with the theoretical analysis.