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

Abstract

Given ν ∈ R + , we consider the following problem: find u > 0,where Ω ⊂ R d , d = 1, 2 or 3, and c > 0 inΩ. We prove H 1 and L ∞ 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 ∈ R + we analyze a more practical scheme involving numerical integration on the nonlinear term. We obtain nearly optimal H 1 and L ∞ error bounds for d = 1. For this case we also present some numerical results. Classification (1991): 65N30, 65N15 Mathematics Subject