Geodesic
Solving the pure Neumann problem by a mixed finite element method
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 25 (2022) no. 4, pp. 385-401

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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. 
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     author = {M. I. Ivanov and I. A. Kremer and Yu. M. Laevsky},
     title = {Solving the pure {Neumann} problem by a mixed finite element method},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
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     publisher = {mathdoc},
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     number = {4},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2022_25_4_a3/}
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M. I. Ivanov; I. A. Kremer; Yu. M. Laevsky. Solving the pure Neumann problem by a mixed finite element method. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 25 (2022) no. 4, pp. 385-401. http://geodesic.mathdoc.fr/item/SJVM_2022_25_4_a3/