Dedicated to J. Tinsley Oden on the occasion of his Sixtieth birthday We present a variant of the Quasi-Minimal Residual (QMR) algorithm of Freund and Nachtigal with a preconditioner based on the Algebraic Multilevel (AMLI) algorithm of Axelsson and Vassil

Multilevel Preconditioned QMR Methods for Unstructured Mesh Computation W. D. Turner, J. E. Flaherty, S. Dey, and M. S. Shephard

Scienti c Computation Research Center (SCOREC) and Department of Computer Science Rensselaer Polytechnic Institute Troy, NY 12180 Dedicated to J. Tinsley Oden on the occasion of his Sixtieth birthday

We present a variant of the Quasi-Minimal Residual (QMR) algorithm of Freund and Nachtigal with a preconditioner based on the Algebraic Multilevel (AMLI) algorithm of Axelsson and Vassilevski. This combination provides an e ective solution method for inde nite algebraic systems and is tested by application to nite element discretizations of the Helmholtz equation. The implementation is applicable with both h- and p- re nements, and easily extends to a parallel environment.

1 Introduction We describe an iterative solution strategy for algebraic systems

Ax= b

(1)

that is intended for use within a framework for large, multi-purpose, exible nite element computation. The framework provides object-oriented tools for the solution of partial di erential systems on both serial and parallel computers. Adaptivity is central and the completed system will provide highlevel support for unstructured mesh computation. Portions of this system involving tetrahedral-element-mesh generation for arbitrary geometries, hand p-re nement, data management, and parallel load balancing are in place 10,11,13,24]. For the breadth of nite element applications, A may be considered as sparse but, not necessarily symmetric or positive de nite. Our approach must be Preprint submitted to Elsevier Preprint 23 January 1997