Rate of convergence for a many point object ( function or surface ).

Question

I am aware of the definition of a Rate of convergence for iterative methods involving a single point $$ x_{n+1} = \phi(x_n) ~~~;~~\lim_{n \rightarrow \infty} {{ |x_{n+1} -\alpha|} \over{|x_{n} -\alpha|^p}} = C $$

However, what definitions and tools we have when working with calculus of variations, trying to converge a function or a surface to some optimal function or surface?

Thanks in advance.