NoteFlux-preserving enforcement of inhomogeneous Dirichlet boundary conditions for strongly divergence-free mixed finite element methods for flow problems

We investigate a flux-preserving enforcement of inhomogeneous Dirichlet boundary conditions for velocity, u|∂Ω=g, for use with finite element methods for incompressible flow problems that strongly enforce mass conservation. Typical enforcement via nodal interpolation is not flux-preserving in general, and it can create divergence error even when divergence-free elements are used. We show with analysis and numerical tests that by slightly (and locally) changing nodal interpolation, the enforcement recovers flux-preservation, is optimally accurate, and delivers divergence-free solutions when used with divergence-free finite elements.