On the Finite Element Solution of the Pure Neumann Problem

This paper considers the finite element approximation and algebraic solution of the pure Neumann problem. Our goal is to present a concise variational framework for the finite element solution of the Neumann problem that focuses on the interplay between the algebraic and variational problems. While many of the results that stem from our analysis are known by some experts, they are seldom derived in a rigorous fashion and remain part of numerical folklore. As a result, this knowledge is not accessible (nor appreciated) by many practitioners—both novices and experts—in one source. Our paper contributes a simple, yet insightful link between the continuous and algebraic variational forms that will prove useful.