Parallel Additive Schwarz Preconditioner for a Discrete Nonlinear Plate Vibration Problem Using a θ-Scheme in Time and Finite Difference in Space

In this paper, we propose a numerical scheme for solving the two-dimensional fourth-order partial differential equation (PDE) with variable coefficients, governing the transverse vibrations of a simply supported thin plate. By introducing a new variable, the equation is transformed into a system of two second-order equations. In the discretization of the spatial derivative, second-order centered finite difference operators are considered, then a three-level $\theta$-scheme is considered for the resulting semi-discretized equations. The stability and convergence of the scheme are proved in the discrete $l_2$ norm and in the discrete maximum norm using the energy method. To accelerate the resolution of the linear system derived from the discretization of the plate equation, the overlapping additive Schwarz preconditioner (ASP) is applied and analyzed. Numerical experiments are provided showing the effectiveness of the preconditioner and the convergence properties of the scheme.