# Non-linear assumed form in Galerkin method

The wikipedia article on Method of mean weighted residuals has a section on choice of test functions, in particular it says about Galerkin method:

The Galerkin method, which uses the basis functions themselves as test functions or in the more general case of a nonlinear assumed form (where the nonlinearity is in the degrees of freedom) of the solution the Galerkin method uses the test functions: $$w_{i}={\frac {\partial u}{\partial a_{i}}}$$

I'm really interested in the "nonlinear assumed form". I would like to know what is known about Galerking when we assume nonlinear form of the trial function.

Unfortunately the wiki article gives only one reference to a book which I was unable to get hold of. Can anyone direct me to any other source where I can read more about this topic?

• Could it be that 'nonlinear assumed form' means that Galerkin function depend nonlinear on degree of freedom $a_i$? This wiki page is indeed very confusing. – daw Dec 21 '17 at 10:23
• I think that is exactly it, but I would like to know more about the consequences of this nonlinearity. All texts talk about Galerkin method where you assume that the unknown function is a linear combination of fixed basis functions. – tom Dec 21 '17 at 21:47