Adaptive methods for semi-linear elliptic equations with critical exponents and interior singularities

Abstract

We consider the effectiveness of adaptive finite element methods for finding the finite element solutions of the parametrised semi-linear elliptic equation Δ u + λ u + u 5 = 0, u > 0, where u ϵ C 2(Ω), for a domain Ω ⊂ R and u = 0 on the boundary of Ω. This equation is important in analysis and it is known that there is a value λ 0 > 0 such that no solutions exist for λ < λ 0 and a singularity forms as λ → λ 0. Furthermore the linear operator L defined by Lφ = Δφ + λφ + 5 u 4φ has a singular inverse in this limit. We demonstrate that conventional adaptive methods (using both static and dynamic regridding) based on usual error estimates fail to give accurate solutions and indeed admit spurious solutions of the differential equation when λ < λ 0. This is directly due to the lack of invertibility of the operator L. In contrast we show that error estimates which take this into account can give answers to any prescribed tolerance.