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Publisher Summary This chapter concentrates on the study that covers the p Galerkin method from a multiscale perspective in the one-dimensional advection- diffiision linear and nonlinear cases and shows the characteristics of the underlying approximate subgrid model. Many methods have been proposed in the literature for stabilizing the plain Galerkin method, the variational multiscale method being one of the most recent and successful. The solution is composed of resolvable and unresolvable scales. The unresolvable scales are smaller than or comparable to the characteristic length scale of the computational grid, and consequently they cannot be accurately represented, as shown by the Nyquist theorem. The hierarchical Galerkin method contains an approximate subgrid model, and it is in this sense stabilized in natural manner. It is shown in the chapter that for p even, the subgrid model is such that the solution will never oscillate for the linear and nonlinear advection-diffusion problem in one dimension. One has also to be aware of the special meaning of the internal degrees of freedom, which should only be used as error estimators.