Adaptive frame methods for nonlinear variational problems

In this paper we develop adaptive numerical solvers for certain nonlinear variational problems. The discretization of the variational problems is done by a suitable frame decomposition of the solution, i.e., a complete, stable, and redundant expansion. The discretization yields an equivalent nonlinear problem on the space of frame coefficients. The discrete problem is then adaptively solved using approximated nested fixed point and Richardson type iterations. We investigate the convergence, stability, and optimal complexity of the scheme. A theoretical advantage, for example, with respect to adaptive finite element schemes is that convergence and complexity results for the latter are usually hard to prove. The use of frames is further motivated by their redundancy, which, at least numerically, has been shown to improve the conditioning of the discretization matrices. Also frames are usually easier to construct than Riesz bases. We present a construction of divergence-free M. Fornasier acknowledges the financial support provided through the IntraEuropean Individual Marie Curie Fellowship Programme, under contract MOIFCT-2006-039438. All of the authors acknowledge the hospitality of Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università di Roma “La Sapienza”, Italy, during the early preparation of this work. The authors want to thank Daniele Boffi, Dorina Mitrea, and Karsten Urban for the helpful and fruitful discussions on divergence-free function spaces. Maria Charina Fachbereich Mathematik, Lehrstuhl VIII, Universität Dortmund Vogelpothsweg 87, 44221 Dortmund, Germany. E-mail: [email protected] Costanza Conti Dipartimento di Energetica “Sergio Stecco”, Università di Firenze, Via C. Lombroso 6/17 50134 Firenze, Italy. E-mail: [email protected] Massimo Fornasier Program in Applied and Computational Mathematics, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544-1000. U.S.A. E-mail: [email protected]