Building upon the fictitious domain approach recently introduced by Gu and Shen [17, 18] for second-order equations, we construct an efficient spectral formulation for solving fourth-order boundary value problems in 2D complex geometries. By embedding irregular regions into a disk, we reduce the two-dimensional fourth-order problem to a series of one-dimensional, fourth-order equations with constant coefficients, which can be solved efficiently by a spectral method, plus a least squares problem, which is essentially one-dimensional, to enforce the boundary conditions. To overcome the ill-conditioning of the least squares problem, a robust Tikhonov-regularized least-squares formulation is used to ensure numerical stability. The method preserves the simplicity of the original differential operator and leads to fast and accurate solvers for fourth-order problems in complex domains. Extensive numerical results demonstrate the fast convergence and broad applicability of our method, showcased by successful solutions to the generalized Stokes problem and the Cahn-Hilliard equation.
