# What are the possible numerical schemes for a diffusion equation with a nonlinear reaction term?

For some simple convex domain $\Omega$ in 2D, we have some $u(x)$ satisfying the following equation: $$-\mathrm{div}(A\nabla u)+cu^n = f$$ with certain Dirichlet and/or Neumann boundary conditions. To my knowledge, applying Newton's method in a finite element space would be a relative straightforward way to numerically solve this equation.

My questions are: (1) Is there a Sobolev theory for the well-posedness of the corresponding variational formulation of this equation assuming zero Dirichlet boundary condition? If so, what Banach space should we consider? (2) What are the possible numerical approaches for this type of equation?

• By "possible numerical approaches", are you asking about discretization or algebraic solvers? Jan 7 '12 at 15:16

• I was assuming that $f$ was a function of space only and not non-linear in $u$. I.e., the only non-linearity in the problem is $u^n$. Jan 7 '12 at 17:44