Operator preconditioning with efficient applications for nonlinear elliptic problems

Nonlinear elliptic partial differential equations form a class of equations that is widespread in modelling various nonlinear phenomena in science, hence their numerical solution has continuously been a subject of extensive research. Such problems also arise from timedependent nonlinear PDE problems, either on the time levels after the time discretization or as describing steady-states of the process. The numerical study of elliptic PDEs has often relied on Hilbert space theory, since one can often incorporate the properties of the continuous PDE problem, from the Hilbert space in which it is posed, into the numerical procedure [3, 11, 13, 14, 25]. An approach that exploits Hilbert space theory is the idea of preconditioning operators, which was developed partly in the author’s papers and then put in general framework in the book [11]. A brief outline is given in [6]. Later this idea was applied again to further nonlinear models, and also coupled with mesh independence results of the author, see [8]. In this survey paper we briefly summarize the main lines of this theory with various applications. The common way of solving a nonlinear PDE is to discretize the problem, which leads to a nonlinear algebraic system normally of very large size, then a suitable iterative solver is applied. A main point here is the choice of suitable preconditioning. The idea of preconditioning operator means that the preconditioner is the projection of a preconditioning operator from the Sobolev space into the same discretization subspace as was used for the original nonlinear problem. In this way the properties of the original problem can be exploited more directly. An important measure of efficiency for an iterative solver is the optimality property, which requires that the computational cost should be of O(n) where n denotes the degrees of freedom in the algebraic system. This holds for some special linear PDE problems. These can then be used as preconditioners to more general problems via the preconditioning operator approach. Then a crucial property of the iteration is