We present a rigorous numerical method to compute global smooth manifolds of solutions of infinite dimensional nonlinear problems. We use a parameter continuation method on a finite dimensional projection to construct a simplicial approximation of the manifold. This simplicial approximation is then used to construct local charts and an atlas of the global manifold in the infinite dimensional space. The idea behind the construction of the smooth charts is to use the so-called radii polynomials to verify the hypotheses of the uniform contraction principle on each simplex. The construction of the manifold is then finalized by proving smoothness along the common lower dimensional faces of adjacent simplices. The method is applied to compute one- and two-dimensional bifurcation manifold of equilibria and time periodic orbits for PDEs.