Sobolev space preconditioning of strogly nonlinear fourth order problems
by
J. Karátson
ELTE University, Budapest, Hungary

Infinite-dimensional gradient method is constructed for nonlinear fourth order elliptic BVPs.

Earlier results on uniformly elliptic equations are extended to strong nonlinearity when the growth conditions are only limited by the Hilbert space well-posedness. The theoretical iteration is executed for the BVP itself (i.e. on the continuous level) in the corresponding Sobolev space, reducing the nonlinear BVP to auxiliary linear (biharmonic) problems. Thus we obtain a class of numerical methods, in which numerical realization is determined by the method chosen for the auxiliary linear problems. The biharmonic operator acts as a Sobolev space preconditioner, yielding a fixed ratio of linear convergence of the iteration (i.e. one determined by the original coefficients only, independent of the way of solution of the auxiliary problems). The approach of the obtained method is opposite to the usual way of first discretizing the problem, on the other hand, it is strongly connected to the Sobolev gradient technique. A numerical example is given for illustration.

Date received: January 26, 2000