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?

  • $\begingroup$ By "possible numerical approaches", are you asking about discretization or algebraic solvers? $\endgroup$
    – Jed Brown
    Jan 7 '12 at 15:16

I see two approaches:

1) Arbitrary f(u). Simply put f ~ f(u0) on the right hand side of the equation, proceed with any non-linear solver, fixed point scheme is a good choice, because you do not have Jacobian anyway. Easiest to implement and use, most general, but possibly inferior performance, because Jacobian can not be exploited (is generally unknown).

2) f(u) decomposed into series (polynomial, Fourier). More difficult to implement and use, may be difficult/impossible for some special f. But in return you can calculate and exploit the Jacobian in a Newton-like method, which will generally result in superior performance.

  • $\begingroup$ 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 $u^n$. $\endgroup$
    – Bill Barth
    Jan 7 '12 at 17:44
  • $\begingroup$ You should add u^n to f. Then you have a simple polynomial form of the reaction term that is best treated with approach 2). $\endgroup$ Jan 7 '12 at 18:44

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