This paper deals with a θ finite difference scheme to solve two-dimensional 4th-order partial differential equation with variable coefficient, which governs the transverse vibrations of a thin plate.First we establish some a priori estimates for the weak solution of our problem.Then, we introduce a new variable, allowing us to convert the plate equation into a system of two second-order differential equations.Stability and convergence analysis are carried out by employing the energy estimate method.We present a class of domain decomposition methods (DDM) called restricted additive Schwarz (RAS). This method is used as a preconditioner for the GMRES algorithm to solve systems of linear equations arising from the discretization of the plate equation by the present scheme.Numerical experiments are provided, confirming the effectiveness of the algorithms.
CHAHID et al. (2026) studied this question.