This thesis is concerned with the continuation theory of incompressible periodic pipe flow. For describing the dynamics of incompressible fluids we use the incompressible Navier-Stokes equation. For a better understanding of it we'll look at its derivation. For a long time now the consensus has been that the laminar solution is linearly stable for all Reynolds numbers. The original idea of this thesis was to adapt a numerical continuation procedure to see if it is possible to jump from the laminar solution branch onto a turbulent one, as it happens in practical experiments. Therefore we inspect the different numerical methods that are used in this procedure. Especially we look at a preconditioner for the linearized problem as the matrix given by the finite element method, using Hood-Taylor elements, becomes less well conditioned as the Reynolds number increases. Prompted by this we look at the convection-diffusion equation and the streamline diffusion discretisation to be able to use it in a multigrid method. To motivate the use of the continuation procedure we look at bifurcation theory, with Fredholm operators and Crandall-Rabinowitz' theorem. We also take a short look at the Allen-Cahn equation to test if the algorithm is correctly defined.