Appending boundary conditions by Lagrange multipliers: Analysis of the LBBcondition

This paper is concerned with the analysis of discretization schemes for sec ond order elliptic boundary value problems when essential boundary conditio ns are enforced with the aid of Lagrange multipliers. Specifically, we show how the validity of the Ladysenskaja-Babuska-Brezzi (LBB) condition for th e corresponding saddle point problems depends on the various ingredients of the involved discretizations. The main result states that the LBB conditio n is satisfied whenever the discretization step length on the boundary, h(G amma) similar to 2(-l), is somewhat bigger than the one on the domain, h(Om ega) similar to 2(-j). This is quantified through constants stemming from t he trace theorem, norm equivalences for the multiplier spaces on the bounda ry, and direct and inverse inequalities. In order to better understand the interplay of these constants, we then specialize the setting to wavelet dis cretizations. In this case the stability criteria can be stated solely in t erms of spectral properties of wavelet representations of the trace operato r. We conclude by illustrating our theoretical findings by some numerical e xperiments. We stress that the results presented here apply to any spatial dimension and to a wide selection of Lagrange multiplier spaces which, in p articular, need not be traces of the trial spaces. However, we do always as sume that a hierarchy of nested trial spaces is given.