Summary. The study of the finite element approximation to nonlinear second order elliptic boundary value problems with mixed Dirichlet-Neumann boundary conditions is presented. In the discretization variational crimes are commited (approximation of the given domain by a polygonal one, numerical integration). With the assumption that the corresponding operator is strongly monotone and Lipschitz-continuous and that the exact solution uEHI(Q) the convergence of the method is proved; under the additional assumption u6HZ(Q), the rate of convergence O(h) is derived without the use of Green's theorem.

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The subject of this paper is the numerical simulation of the interaction of two-dimensional incompressible viscous flow and a vibrating airfoil. A solid airfoil with two degrees of freedom, which can rotate around the elastic axis and oscillate in the vertical direction, is considered. The numerical simulation consists of the finite element solution of the Navier-Stokes equations, coupled with the system of ordinary differential equations describing the airfoil motion. The high Reynolds numbers considered 10 5 210 6 require the application of a suitable stabilization of the finite element discretization. The method presented in this paper is based on the laminar model and the turbulence modelling is not applied here. The time-dependent computational domain and a moving grid are taken into account with the aid of the arbitrary Lagrangian-Eulerian (ALE) formulation of the Navier-Stokes equations. Special attention is paid to the time discretization and the solution of the nonlinear discrete problem on each time level is performed. As a result, a sufficiently accurate and robust method is developed, which is applied to the case of flow-induced airfoil vibrations with large amplitudes after the loss of aeroelastic stability. The computational results are compared with known aerodynamical data and with results of aeroelastic calculations obtained by NASTRAN code for a linear approximation. r