Two-Sided Approximation Method Using Green’s Function and Green – Rvachev’s Quasi-Function in the Numerical Analysis of Mems Governed by the Navier Problem

In modern technologies, various microelectromechanical systems are widely used. These systems are integrated devices that combine mechanical elements, sensors, actuators, and electronics at the microscale. Mathematical modeling is extensively employed in the design of such systems; therefore, the scientific problem of developing new and improving existing numerical methods for the analysis of corresponding boundary value problems is of significant relevance. This paper considers a planar Navier problem for a fourth-order semilinear elliptic equation, which serves as a mathematical model of a microelectromechanical system consisting of a thin elastic deformable plate located above a rigid, immobile grounded electrode, with the plate hinged along its boundary. For the numerical analysis of the considered boundary value problem, a reduction to a system of second-order semilinear elliptic equations is performed, and two iterative schemes of the method of two-sided approximations are proposed. The first scheme is based on the use of the classical Green’s function and the transformation of the boundary value problem into an equivalent system of Hammerstein integralequations. The second scheme employs the Green-Rvachev’s quasi-function, constructed using the constructive apparatus of the theory of R  functions,to build an equivalent system of Urysohn integral equations. Both systems of integral equations are analyzed using methods of nonlinear analysisin semi-ordered Banach spaces. Conditions for the convergence of the constructed iterative processes are obtained, as well as sufficient conditions forthe existence of a unique (on a certain conical segment) positive solution to the original boundary value problem. The efficiency of the proposed iterative schemes is demonstrated by computational experiments for test parameter values in the case of a plate shaped as a unit square. It is noted that theapplication of the method of two-sided approximations has several advantages, the most important of which is the availability of a convenient a posteriori error estimate, allowing one to control the accuracy of approximation at each iteration.