A nontraditional approach for solving the Neumann problem by the finite element method. (English) Zbl 0771.65070

A new variational formulation of a second order elliptic problem with Neumann boundary conditions is described. The formulation does not require any quotient spaces and is advisable for finite element approximations.
It is based on the generalized Poincaré inequality: \(\| v\|_ 1\leq C\left(| v|^ 2_ 1+\left(\int_ \omega v d\omega\right)^ 2\right)^{1/2}\) \(\forall v\in H^ 1(\Omega)\), where \(\| v\|_ 1\) and \(| v|_ 1\) are, respectively, the standard norm and seminorm in \(H^ 1(\Omega)\), and \(\omega\) is only an open set either in \(\Omega\) or in \(\Gamma_ 0\), where \(\Gamma_ 0\) is the boundary of a domain \(\Omega_ 0\subseteq\Omega\).


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations