A novel mixed spectral method with ball polynomials for the Biharmonic equation on a unit ball

A novel mixed spectral-Galerkin method based on generalized ball polynomials is proposed for solving the biharmonic equation on a unit ball. By introducing an auxiliary variable to decouple the biharmonic equation into a system of second-order equations, the corresponding discrete scheme yields a strictly diagonal stiffness matrix, which significantly enhances the computational efficiency. Rigorous a-priori error estimates are established to demonstrate the exponential convergence rates in both the $L^2$- and $H^1$-norms. Extensive numerical experiments are conducted to verify the theoretical analysis and confirm the high efficiency and accuracy of the proposed scheme.