Finite element approximation of a semilinear elliptic problem with a singular nonlinearity

Abstract.

Given $\nu \in{\Bbb R}_{+}$, we consider the following problem: find $u > 0$, such that \begin{displaymath} -\Delta u = cu^{-\nu} \quad \mbox{in}\,\, \Omega\,, \quad u = 0 \quad \mbox{on}\,\, \partial \Omega\,, \end{displaymath} where $\Omega \subset{\Bbb R}^{d},\, d = 1,\,2$ or 3, and $c > 0$ in $\bar{\Omega}$. We prove $H^{1}$ and $L^{\infty}$ error bounds for the standard continuous piecewise linear Galerkin finite element approximation with a (weakly) acute triangulation. Our bounds are nearly optimal. In addition, for d = 1 and 2 and $c \in{\Bbb R}_{+}$ we analyze a more practical scheme involving numerical integration on the nonlinear term. We obtain nearly optimal $H^{1}$ and $L^{\infty}$ error bounds for d = 1. For this case we also present some numerical results.