A multipoint finite-difference scheme for the problem of bending of rectangular orthotropic plates with freely supported edges: Construction and convergence estimate

A. P. Kubanskaya

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms, Tome 80 (1978), pp. 66-82

Résumé

The boundary-value problem \begin{gather*} D_1\dfrac{\partial^4w}{\partial x^4}+2D_2\dfrac{\partial^4w}{\partial x^2\partial y^2}+D_3\dfrac{\partial^4w}{\partial y^4}=f(x,y) \\ W|_{y=0;b}=0,\quad\dfrac{\partial^2w}{\partial y^2}|_{y=0'b}=0;\quad W|_{x=-a;a}=0,\quad \dfrac{\partial^2w}{\partial y^2}|_{x=-a'a}=0 \end{gather*} of static deflection of a rectangular orthotropic plate is replaced with a finite-difference problem. The rectangle $[-a\leqslant x\leqslant a, 0\leqslant y\leqslant b]$ is partitioned into a mesh with step $h$ in the direction $y$ and $h_1$, in the direction $x$; second derivatives with respect to $y$ and $x$ are replaced with multipoint approximations using the templates $2p+1$, $2p_1+1$ (where $p$ and $p_1$ are arbitrary natural numbers) with errors $O(h^{2p})$, $O(h^{2p_1})$; the fourth-order derivatives are replaced with approximations using the templates $4p+1$ and $4p_1+1$ with the same errors. The finite-difference system of linear algebraic equations is transformed into a decomposable system. The convergence of the proposed method is estimated.