The Sobolev gradient concept has been developed in the 1970s, with a first publication in 1985, and an introduction can be found at: [Ranka][1]

I would like to learn how strong the impact of Sobolev gradients has been in numerical partial differential equations.

Is it one of several possible tools for nonlinear pde, or a powerful approach without "competitors"?


  [1]: http://www.cse.unt.edu/~renka/papers/sobolev.pdf%20%22Paper%20by%20Ranka%22%22