On finite element discretizations of the pure Neumann problem

We develop a general variational framework for the finite element solution of the pure Neumann problem. Our starting point is the equivalence between the weak Neumann equation and a minimization problem on the factor space H1(Ω)/RI . This problem gives rise to a family of minimization problems posed on conventional Sobolev spaces but constrained by a generalized zero-mean condition. We show that the constraint’s choice and implementation is the paradigm that describes finite element methods for the Neumann problem. In terms of finite element algebraic problems it specializes to a discrete projection choice and reveals connections with methods for equality constrained quadratic programs. We use this paradigm to develop a novel class of regularized finite elements. Numerical experiments indicate that this class consistently outperforms other finite elements for the Neumann problem.