A novel numerical method for simulating time-dependent flow of viscoelastic fluids derived from dumbbell models is described. The constitutive equation is solved in a co-deforming frame, where the natural time-derivative is the upper-convected derivative. Mesh reconnection is achieved using a variant of Delaunay triangulation. The velocity and pressure are found via a finite element solution of the momentum equations. The method is tested by applying it to the benchmark problem of a sphere falling along the axis of a cylindrical tube.