Narrow Results

We consider the buckling eigenvalue problem for a clamped plate in the annulus. We identify the first eigenvalue in dependence of the inner radius, and study the number of nodal domains of the corresponding eigenfunctions. Moreover, in order to investigate the asymptotic behavior of eigenvalues and eigenfunctions as the inner radius approaches the outer one, we provide an analytical study of the buckling problem in rectangles with mixed boundary conditions.

The aim of this paper is to simulate the two-dimensional stationary Stokes problem. In vorticity-Stream function formulation, the Stokes problem is reduced to a biharmonic one; this approach leads to a formulation only based on the stream functions and therefore can only be applied to two-dimensional problems. The idea developed in this paper is to use the discretization of the Laplace operator by the nonconforming $P1$ finite element. For the solutions which admit a regularity greater than $H_0^2(\mathrm{\Omega })$, in the general case, the convergence of the method is shown with the techniques of compactness. For solutions in $C^4(\bar{\mathrm{\Omega }}),$ an error estimate is proved, and numerical experiments are performed for the steady-driven cavity problem.