Multilevel Preconditioners for Lagrange Multipliers in Domain Imbedding

Janne Martikainen
Department of Mathematical Information Technology
University of Jyvaskyla
P.O. Box 35 (Agora)
FIN-40351 Jyvaskyla
Finland

Tuomo Rossi
Department of Mathematical Information Technology
University of Jyvaskyla
P.O. Box 35 (Agora)
FIN-40351 Jyvaskyla
Finland

Jari Toivanen
Department of Mathematical Information Technology
University of Jyvaskyla
P.O. Box 35 (Agora)
FIN-40351 Jyvaskyla
Finland


Abstract

A domain imbedding method where the Dirichlet boundary conditions are treated using boundary supported Lagrange multipliers isconsidered. The discretization leads to a saddle-point problem which is solved iteratively by using either the PMINRES method with a block-diagonal preconditioner or the PCG method in an Uzawa type approach. In both cases, the preconditioning of the Schur complement related to Lagrange multipliers is based on a special sparse implementationof BPX/MDS method. The developed preconditioning technique is well-suited even for three-dimensional problems in domains with complicated shapes. Several numerical experiments for two-dimensional and three-dimensional problems demonstrate the efflciency and the applicability of the proposed method.

Key words. domain imbedding method, Lagrange multipliers, multilevel methods, preconditioning

AMS subject classifications. 65F10, 65N22, 65N55