Appending boundary conditions by Lagrange multipliers: Analysis of 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. Specifically, we show how the validity of the Ladyškaja–Babušska–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 satisfied whenever the discretization step length on the boundary, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $h_\Gamma \sim 2^{-\ell}$\end{document}, is somewhat bigger than the one on the domain, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $h_\Omega \sim 2^{-j}$\end{document}. This is quantified through constants stemming from the trace theorem, norm equivalences for the multiplier spaces on the boundary, and direct and inverse inequalities. In order to better understand the interplay of these constants, we then specialize the setting to wavelet discretizations. In this case the stability criteria can be stated solely in terms of spectral properties of wavelet representations of the trace operator. We conclude by illustrating our theoretical findings by some numerical experiments. 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. However, we do always assume that a hierarchy of nested trial spaces is given.