A mesh method for solving fourth-order quasilinear elliptic equations

Abstract

A mixed finite element method for solving the Dirichlet problem for a fourth-order quasilinear elliptic equation in divergent form was proposed and investigated. It was assumed that the domain in which the problem is solved is bounded and has a dimension greater or equal to two. When constructing the finite element scheme, all the second derivatives of the required solution were chosen as auxiliary unknowns. The usual triangulation of the domain by Lagrangian simplicial (triangular) elements of orders two and higher was used. Under the assumption that the operator of the original problem satisfies the standard conditions of bounded nonlinearity and coercivity, the existence of an approximate solution for any value of the discretization parameter was proved. The uniqueness of the approximate solution was established under tighter restrictions, namely, assuming the Lipschitz-continuity and the strong monotony of the differential operator. Under the same conditions, a two-layer iterative process was constructed, and the estimation of the convergence rate independent of the discretization parameter was proved. Accuracy estimates for the approximate solutions, optimal in the case of linearity of the differential equation, were obtained. The results of the application of the proposed technique to the problem of the equilibrium of a thin elastic plate were presented.

Keywords

• mixed finite element method
• accuracy estimates
• iterative method
• convergence rate estimates
• theory of plates