We consider a biharmonic problem ${\Delta }^2{u}_{\omega }=f_{\omega }$ with Navier type boundary conditions $u_{\omega }=\Delta u_{\omega }=0$, on a family of truncated sectors ${\Omega }_{\omega }$ in ${ℝ}^2$ of radius $r$, $0<r<1$ and opening angle $\omega$, $\omega \in (2\pi /3,\pi ]$ when $\omega$ is close to $\pi$. The family of right-hand sides ${\left(f_{\omega }\right)}_{\omega \in (2\pi /3,\pi ]}$ is assumed to depend smoothly on $\omega$ in $L^2\left({\Omega }_{\omega }\right)$. The main result is that $u_{\omega }$ converges to $u_{\pi }$ when $\omega \to \pi$ with respect to the $H^2$-norm. We can also show that the $H^2$-topology is optimal for such a convergence result.

Let $A$ be the $L^p$ realization ($1\&lt;p\&lt;\infty$) of a differential operator $P(D_x,D_t)$ on ${ℝ}^n\times {ℝ}^{+}$ with general boundary conditions $B_k(D_x,D_t)u(x,0)=0$ ($1\le k\le m$). Here $P$ is a homogeneous polynomial of order $2m$ in $n+1$ complex variables that satisfies a suitable ellipticity condition, and for $1\le k\le m$$B_k$ is a homogeneous polynomial of order...

A conformal finite element method is investigated for a dual variational formulation of the biharmonic problem with mixed boundary conditions on domains with piecewise smooth curved boundary. Thus in the problem of elastic plate the bending moments are calculated directly. For the construction of finite elements a vector potential is used together with $C^0$-elements. The convergence of the method is proved and an algorithm described.

Estimates for the combined effect of boundary approximation and numerical integration on the approximation of (simple) eigenvalues and eigenvectors of 4th order eigenvalue problems with variable/constant coefficients in convex domains with curved boundary by an isoparametric mixed finite element method, which, in the particular case of bending problems of aniso-/ortho-/isotropic plates with variable/constant thickness, gives a simultaneous approximation to bending moment tensor field $\Psi ={\left({\psi }_{ij}\right)}_{1\le i,j\le 2}$ and displacement...

Estimates for the combined effect of boundary approximation and numerical integration on the approximation of (simple) eigenvalues and eigenvectors of 4th order eigenvalue problems with variable/constant coefficients in convex domains with curved boundary by an isoparametric mixed finite element method, which, in the particular case of bending problems of aniso-/ortho-/isotropic plates with variable/constant thickness, gives a simultaneous approximation to bending moment tensor field $\Psi ={\left({\psi }_{ij}\right)}_{1\le i,j\le 2}$ and displacement...

In this paper, the Babuška’s theory of Lagrange multipliers is extended to higher order elliptic Dirichlet problems. The resulting variational formulation provides an efficient numerical squeme in meshless methods for the approximation of elliptic problems with essential boundary conditions.

In this paper, the Babuška's theory of Lagrange multipliers is extended to higher order elliptic Dirichlet problems. The resulting variational formulation provides an efficient numerical squeme in meshless methods for the approximation of elliptic problems with essential boundary conditions.