Multilevel preconditioning — Appending boundary conditions by Lagrange multipliers

For saddle point problems stemming from appending essential boundary conditions in connection with Galerkin methods for elliptic boundary value problems, a class of multilevel preconditioners is developed. The estimates are based on the characterization of Sobolev spaces on the underlying domain and its boundary in terms of weighted sequence norms relative to corresponding multilevel expansions. The results indicate how the various ingredients of a typical multilevel framework affect the growth rate of the condition numbers. In particular, it is shown how to realize even condition numbers that are uniformly bounded independently of the discretization.These investigations are motivated by the idea of employing nested refinable shift-invariant spaces as trial spaces covering various types of wavelets that are of advantage for the solution of boundary value problems from other points of view. Instead of incorporating the boundary conditions into the approximation spaces in the Galerkin formulation, they are appended by means of Lagrange multipliers leading to a saddle point problem.

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