A novel BDF-spectral method and its error analysis for Cahn–Hilliard equation in polar geometry

In this paper, we first propose and study a finite difference Legendre-Fourier spectral method for solving the Cahn–Hilliard equation in polar geometry. The fundamental idea is to restate the original problem in an equivalent form under polar coordinates. Subsequently, by introducing an auxiliary second-order equation, we transform it into a coupled second-order nonlinear system. Furthermore, we introduce a class of weighted Sobolev spaces and their approximation spaces, formulate first- and second-order semi-implicit schemes for the coupled second-order nonlinear system, and demonstrate the stability of these schemes under specific conditions on the time step. In particular, the introduction of pole singularities and the nonlinearity of coupling problem pose significant challenges to theoretical analysis. To overcome these difficulties, we construct a new class of projection operators and prove their approximation properties, thereby providing error estimates for the approximate solutions. Finally, we provide numerous numerical examples, and the numerical results confirm the effectiveness of the algorithm and the correctness of the theoretical findings.