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Local stability conditions for the Babuška method of Lagrange multipliers

Author: Juhani Pitkäranta
Journal: Math. Comp. 35 (1980), 1113-1129
MSC: Primary 65N30
MathSciNet review: 583490
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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.

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Article copyright: © Copyright 1980 American Mathematical Society