Treating Inhomogeneous Essential Boundary Conditions in Finite Element Methods and the Calculation of Boundary Stresses

Finite element approximations of the Stokes and Navier–Stokes equations with in-homogeneous essential boundary conditions are considered. Boundary conditions are enforced weakly by introducing Lagrange multipliers. Optimal error estimates, including some for the stress vector on the boundary, are derived under minimal regularity assumptions on the data. Particular attention is paid to the analysis of a practical choice of finite element spaces for which the Lagrange multiplier calculation uncouples from that for the velocity and pressure. The results are also applicable to general second-order elliptic systems.