Boundary value problems with discontinuities and monotone discretization
The aim of the paper is to provide a constructive and numerically implementable procedure for calculating lower and upper solutions of the following two-point boundary value problem \(-u''(x) = h(u(x) - a) p(u(x))\), for any \(x \in (0,1)\), \(u(0) = u(1) = 0\), where \(h\) denotes the Heaviside function and \(p : \mathbb{R} \to \mathbb{R}_ +\), is an arbitrary continuous function such that some \(\sigma \geq 0\) exists satisfying \(p(s) + \sigma s \leq p(t) + \sigma t\), for any \(0 \leq s \leq t\). Furthermore, \(a > 0\) is some given constant. The authors introduce iterations on nested grids instead of approximating the problem on a fixed grid, and prove the related monotone iteration discretization technique to converge even in the case of discontinuous right-hand sides, despite classical linearization techniques can fail to converge in the situation under consideration.
scientific article; zbMATH DE number 699116
scientific article