Appending Boundary Conditions by Lagrange Multipliers: General Criteria for the Lbb Condition

This paper is concerned with the analysis of discretization schemes for second order elliptic boundary value problems when essential boundary conditions are enforced with the aid of Lagrange multipliers. Speciically, we show how the validity of the Lady senskaja{Babu ska{Brezzi (LBB) condition for the corresponding saddle point problems depends on the various ingredients of the involved discretizations. The main result states that the LBB condition is satissed whenever the discretization step length on the boundary, h ? 2 ?` , is somewhat bigger than the one on the domain, h 2 ?j. This is quantiied through constants stemming from the trace theorem, norm equivalences for the multiplier spaces on the boundary, and direct and inverse inequalities. We stress that the results presented here apply to any spatial dimension and to a wide selection of Lagrange multiplier spaces which, in particular, need not be traces of the trial spaces.