Existence and error estimates for solutions of a discrete analog of nonlinear eigenvalue problems

Abstract:

Finite-difference methods using the five-point discrete Laplacian and suitable boundary modifications for approximating $(1) - \Delta u = \lambda f(x,u)$ in a plane domain $D,u = 0$ on its boundary are considered. It is shown that if (1) has an isolated solution, $u$, then the discrete problem has a solution, ${U_h}$, for which ${U_h} - u = O({h^2})$. If the discrete problem has solutions, ${U_h}$, such that $|{U_h}| \leqq M$ as $h$ tends to zero, then (1) has a solution, $u$, satisfying $|u| \leqq M$. Let ${\lambda ^\ast }$ be a critical value of $\lambda$ so that (1) has positive solutions for $\lambda \leqq \lambda ^\ast$ but not for $\lambda > {\lambda ^\ast }$, then the discrete problem has an analogous critical value $\lambda _h^\ast$ and, under suitable conditions, $\lambda _h^\ast - {\lambda ^\ast } = O({h^{4/3 - \epsilon }}),\epsilon > 0$. Computed results for the case $f(x,u) = {e^u}$ and $D$ the unit square are given.