Thanks for your answer Hauke.  The Poisson equation should be "well posed" as the equilibrium state should be reached when the source flux is matched by the fluxes at the boundaries, right.  Physically, the diffused quantity at the boundaries would then go to to a virtual reservoir of infinite size.

 As noted by Hauke, in the FEM formulation, one either specifies Dirichlet conditions or generalized Neumann ( aka Robin) conditions at the boundary. If you would like to see some examples of this check out the following attached PDF ( using Mathematica). Other  examples including Mathematica notebooks can be down loaded from the following site:  https://sites.google.com/site/ekayasolutionsdiffeqns/home/finite-element-method/mathematica

Please do tell us what exactly your boundary conditions look like. Then we can have a better idea. Then what about the heat load, is concentrated?or equal distributed accross the space? What about your system? Cartesian?Cylindrical? Sometimes in every eqaution some terms that are troublesome can be neglected or approximated with other terms,far easier, to be solved. Then you would have your first idea of how your solution look likes. Then you can start to approximate it even better.

Lastly what have personally helped me, was to approximate roughly the non linear terms, and then move them to the right side of the equation. Then this means that they act as a second heat load. Then you can solve your problem easily, and play around with your assumptions, in order to understand your problem and finally solve it.

Hope it helps.

Dear David,

There is no direct answer to this, as it depends on the problem and on the numerical scheme you are using. 

I think the correct question is whether the weak formulation of your problem has a solution and how many (if not unique). For this you have to define the proper Hilbert spaces. If the boundary conditions are of homogeneous (in this case = 0) Dirichlet type, then you choose the space of functions having a vanishing trace on the boundary. If you have for example Neumann or Robin type of data on the entire boundary, or on a part of it, these are automatically included through the weak formulation. Take care, you may need to check the compatibility condition in the former case. 

Once you have a positive answer to the existence/uniqueness question, you can continue by approximating the space by a finite dimensional one. In this sense you can consider a finite element space to find a numerical solution in a finite dimensional space. In doing so, you formulate a discrete version of the problem in the weak form. The you may need to solve some nonlinear algebraic systems, unless you formulate the discretized problem in such a way that it is linear. By refining the mesh, the approximation converges hopefully to the weak solution. 

Sorry for the technical answer, I was trying to explain why one cannot give a general yes/no answer. The more specific you are, the better the chances to receive a suitable answer. 

All the best,

Sorin

I think I understand what you say. while people and mathematics say this case be solved, it has some problem especially in nonlinear case. I encounter same problem in numerical solution of  nonlinear wave equation! It seems that nonlinear case is sensitive to non-Dirichlet b.c.  

Dear

I agree with Sorin answer

regards