Finite Element Method Solutions for Semi - linear

We compute general (non-radial) positive solutions of the semi-linear elliptic PDE u + u + u 5 = 0, in 3 space dimensions (where the nonlinearity is critical) using the Finite Element Method. We have overcome two fundamental diiculties in this approach. Firstly the convergence of the numerical solutions is very slow (on regular grids) and it is necessary to work with very ne meshes (10 6 nodes). A full Newton iteration is then infeasible, and we have succesfully implemented an approximate Newton method which does not require storage of the full Jacobian, but nevertheless still gives very fast convergence, even when the solution is developing a `spike.' Secondly, although there is a critical value 0 such that there is no positive solution of the PDE for < 0 , the FEM discretization is nite dimensional and variational theory then shows that there exists a numerical solution for all < 1. However using formal asymptotics we can recover information about the solution of the PDE from the numerics and use this to approximate 0 , in particular obtaining numerical evidence that corroborates McLeod's conjecture, that 0 = , in the case is a cube.