Adaptive Methods for Semi-linear Elliptic Equations with Critical Exponents and Interior Singularities

We consider the eeectiveness adaptive nite-element methods for nding the nite element solutions of the parametrised semi-linear elliptic equation u+u+u 5 = 0; u > 0, where u 2 C 2 ((); for a domain IR 3 and u = 0 on the boundary of : This equation is important in analysis and it is known that there is a value 0 > 0 such that no solutions exist for < 0 and a singularity forms as ! 0. Furthermore the linear operator L deened by L' = '+'+5u 4 ' 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 diierential equation when < 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. Adaptive methods for semi-linear elliptic equations with singularities.