Numerical methods for fourth-order nonlinear elliptic boundary value problems (Q2747810)

The purpose of the article is to construct three monotone iterative schemes for the computation of the solution of a finite-difference scheme that approximates the solution of a fourth-order elliptic boundary value problem in the form NEWLINE\[NEWLINE\Delta (b(x)\Delta u)=f(x, u, \Delta u),\;x \in \Omega,\;B[u]=g(x),\;B[b\Delta u]=g^{*} (x),\;x \in \partial\Omega.NEWLINE\]NEWLINE Here \(\Omega\) is a bounded domain in \(R^n\) with boundary \(\partial \Omega, \Delta\) is the Laplace operator, and \(B[w]=\alpha {\partial w}/{\partial v} + \beta(x) w\) with \(\partial /{\partial v}\) denoting the outward normal derivative on \(\partial \Omega\). Each of these schemes yields two sequences that converge monotonically to a maximal and a minimal solutions, respectively, of the finite difference scheme. The main requirement are the nondecreasing or nonincreasing property of \(f(x, w, v)\) in \(u\) and the existence of a pair of upper and lower solutions.