Nonlinear diffusion problems. Existence and uniqueness of the solution

Abstract

The elliptic type differential equation is the mathematical model for steady-state diffusion problems. In certain engineering cases, the diffusion coefficient may not be constant, for example, depending on concentration or temperature. This leads to nonlinear boundary value problems whose analytical solution is generally complex and very laborious to obtain. On the other hand, the most important thing before addressing the resolution of this type of problem is to determine the existence of the solution and, in that case, the conditions under which its uniqueness can be ensured. In the present work, starting from the weak formulation of the boundary value problem, the theoretical basis of the finite element method, and useful for its implementation in computers, the continuity and coercivity of the nonlinear form associated with the boundary value problem are demonstrated. With these results, the Sobolev space to which the solution belongs is established, and the conditions for the existence and uniqueness of the solution are determined. Finally, the numerical results obtained using the finite element method are validated by comparing them with solutions available in the literature.