VARIABLE PRECONDITIONING VIA QUASI-NEWTON METHODS FOR NONLINEAR PROBLEMS IN HILBERT SPACE ∗ J ´ ANOS KAR ´ ATSON † AND ISTV ´ AN FARAG ´ O † SIAM J. NUMER. ANAL. c  2003 Society for Industrial and Applied Mathematics Vol. 41, No. 4, pp. 1242–1262 Abstract. The aim of this paper is to develop stepwise variable preconditioning for the iterative solution of monotone operator equations in Hilbert space and apply it to nonlinear elliptic problems. The paper is built up to reflect the common character of preconditioned simple iterations and quasi- Newton methods. The main feature of the results is that the preconditioners are chosen via spectral equivalence. The latter can be executed in the corresponding Sobolev space in the case of elliptic problems, which helps both the construction and convergence analysis of preconditioners. This is illustrated by an example of a preconditioner using suitable domain decomposition. Key words. variable preconditioning, quasi-Newton methods, iterative methods in Hilbert space, nonlinear elliptic problems AMS subject classifications. 35J65, 65J15 DOI. 10.1137/S0036142901384277 1. Introduction. The aim of this paper is to develop stepwise variable precon- ditioning for the iterative solution of monotone operator equations F (u)=0 (1) in Hilbert space and apply it to nonlinear elliptic boundary value problems. Nonlinear elliptic problems arise in many applications in physics and other fields, for instance in elastoplasticity, magnetic potential equations, and flow problems. The most frequently used numerical methods for nonlinear elliptic problems rely on some discretized form of the problem, whose solution is obtained by an iterative method. Simpleiterationisoftenabletoyieldfavorablespeedofglobalconvergenceifsupplied with suitable preconditioning, and in these cases its usage can be justified versus Newton’s method owing to the extra work of forming the Jacobians (see, e.g., [2, 5]). Hence,similarlytolinearproblems,preconditioningismosttimesacrucialelementof theconstructionoftheiterativemethod. Thechoiceofpreconditionersisoftenhelped by Hilbert space background, which helps both the construction of methods and the study of convergence. A typical example of this is the Sobolev gradient technique [25,26]. (Fortheauthors’relatedresultssee,e.g.,[6,15,20].) Inthecaseofmonotone operators a natural kind of preconditioning is based on spectral equivalence, in an analogous way to symmetric linear equations. Namely, preconditioners are chosen to begloballyspectrallyequivalenttothederivatives F ′ (u)oftheoperatorineachpoint. AHilbertspaceframeworkhasbeendevelopedforthisin[21],inwhichpreconditioners for the discretized elliptic systems are found as projections of linear operators chosen as preconditioners for the original nonlinear differential operator. The above described simple iterations are globally preconditioned in the sense that the preconditioners are the same in each step and rely on the global behavior of ∗ Received by the editors January 29, 2001; accepted for publication (in revised form) January 14, 2003; published electronically August 22, 2003. This research was supported by the Hungarian National Research Funds AMFK under Magyary Zolt´an Scholarship and by OTKA under grants F022228 and T031807. http://www.siam.org/journals/sinum/41-4/38427.html † Department of Applied Analysis, ELTE University, H-1518 Budapest, Hungary (karatson@cs. elte.hu, faragois@cs.elte.hu). 1242