Local stability conditions for the Babuška method of Lagrange multipliers

Abstract:

We consider the so-called Babuška method of finite elements with Lagrange multipliers for numerically solving the problem $\Delta u = f$ in $\Omega$, $u = g$ on $\partial \Omega$, $\Omega \subset {R^n}$, $n \geqslant 2$. We state a number of local conditions from which we prove the uniform stability of the Lagrange multiplier method in terms of a weighted, mesh-dependent norm. The stability conditions given weaken the conditions known so far and allow mesh refinements on the boundary. As an application, we introduce a class of finite element schemes, for which the stability conditions are satisfied, and we show that the convergence rate of these schemes is of optimal order.