A review on some discrete variational techniques for the approximation of essential boundary conditions
We review different techniques to enforce essential boundary conditions, such as the (nonhomogeneous) Dirichlet boundary condition, within a discrete variational frame- work, and especially techniques that allow to account for them in a weak sense. Those are of special interest for discretizations such as geometrically unfitted finite elements or high order methods, for instance. Some of them remain primal, and add extra terms in the discrete weak form without adding a new unknown: this is the case of the boundary penalty and Nitsche techniques. Others are mixed, and involve a Lagrange multiplier with or without stabilization terms. For a simple setting, we detail the different as- sociated formulations, and recall what is known about their stability and convergence properties.