A solution of the degenerate Neumann problem by the finite element method

This paper deals with the solution of the degenerate Neumann problem for the diffusion equation by the finite element method. First, an extended generalized formulation of the Neumann problem in the Sobolev space $H^1(\Omega)$ is derived and investigated. Then a discrete analogue of this problem is formulated using standard finite element approximations of the space $H^1(\Omega)$. An iterative method for solving the corresponding SLAE is proposed. Some examples of solving the model problems are used to discuss the numerical peculiarities of the algorithm proposed.