Solving the pure Neumann problem by a mixed finite element method

This paper proposes a new method for the numerical solution of a pure Neumann problem for the diffusion equation in a mixed formulation. The method is based on the inclusion of a condition of unique solvability of the problem in one of the equations of the system with a subsequent decrease in its order by using a Lagrange multiplier. The unique solvability of the problem obtained and its equivalence to the original mixed formulation in a subspace are proved. The problem is approximated on the basis of a mixed finite element method. The unique solvability of the resulting saddle system of linear algebraic equations is investigated. Theoretical results are illustrated by computational experiments.