Convergence of Finite Element Approximations for Generalized Marguerre–von Kármán Equations

Abstract

In this work, we establish the convergence of a conforming finite element approximations to the generalized Marguerre–von Karman equations. More precisely, we consider here the generalized Marguerre–von Karman equations, which constitute a mathematical model for a nonlinearly elastic shallow shell subjected to boundary conditions of von Karman’s type only on a portion of its lateral face, the remaining portion being free. We first reduce the discrete problem of these equations to a single discrete cubic operator equation, whose unknown is the approximate of vertical displacement of the shallow shell. We next solve this discrete operator equation, by adapting a compactness method due to J.L. Lions and on Brouwer’s fixed point theorem (Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod, Paris, 1969). Then we establish the convergence of a conforming finite element approximations to these equations.