we have the following problem: $$ u_t +u u_x - \nu u_{xx} = f(x,t); \ x \in \mathbb{R}, t \in [0,T] $$ where $\nu > 0$, with the initial condition: $$ u(x,0)= g(x), $$ and boundary conditions: $$ u(0,t)= r_1(x), \ u(1,t)=r_2(x). $$
I want to apply finite difference on time and finite elements $P_2$ in space to discretize this problem and obtain an aproximate solution.
I try this:
Firts, we linearize the problem using a simple fixed point method. We introduce a $nith =0$ and consider the linear problem
$$ u_t +nith * u_x - \nu u_{xx} = f(x,t); \ x \in \mathbb{R}, t \in [0,T] $$ then we discritize this problem in time and space, we obtain $u_h$ and we pu $nith=u_h$ then we repete the discretisation.
Concerning the discretization, we discretize in time, then we write $$ \partial_t u_h = \dfrac{u_h - U_\mathrm{old}}{dt}, $$ where $dt$ is the step time discretisation. My questions are:
- What's the variational formulation associated to this problem?
- What's the space of approximation using to apply the finite elements? And what's the linear problem $K_h u_h =b_h$ associated?
Best regards.