We consider the following equation with boundary and initial conditions:
$\varrho(u)u_t = \frac{\partial}{\partial x}\left(\alpha(u)\frac{\partial u}{\partial x}\right),\quad x\in (0,1)$
$u(0,t) = 0$
$u(1,t) = 0$
$u(x,0) = I(x)$
There is two nonlinearities in this equation that needs to be treated when we solve the equation.
The starting point of most of the following derivations will be the residual:
$$ R(u) = \varrho(u)u_t - \frac{\partial}{\partial x} (\alpha(u)\frac{\partial u}{\partial x}) $$If we discretize in time with a backward euler scheme and then use a Galerkin discretization for the spatial part, we find
$$ \varrho(\Delta t n)u_t(\Delta t n) \approx [\varrho(u) D_t^- u]^n = \frac{\varrho(u^n)}{\Delta t}(u^{(n)}-u^{(n-1)}) $$$$ u^n \approx \sum_i c_i^n \phi_i(x) $$Where the basis is spanned by a set of $\phi_i(x)$´s.
We demand the solution to be valid in the space spanned by the basis functions: $v = span ({\phi_i})$
$$(R,v) = (R(c_0,c_1, ...), v) = 0$$This leads to a linear system of equations with $u^{(n)}(x,c_0,c_1,...)$ unknown:
$$ (R,v) = (\frac{\varrho(u^n)}{\Delta t}(u^{(n)}-u^{(n-1)}), v ) - (\frac{\partial}{\partial x} (\alpha(u^{(n)})\frac{\partial}{\partial x} u^{(n)}), v) = 0 $$