Sobolev gradients: a nonlinear equivalent operator theory in preconditioned numerical methods for elliptic PDEs

Solution methods for nonlinear boundary value problems form one of the most important topics in applied mathematics and, similarly to linear equations, preconditioned iterative methods are the most efficient tools to solve such problems. For linear equations, the theory of equivalent operators in Hilbert space has proved an efficient organized framework for the study of preconditioners [6, 9], in particular when one looks for mesh independent convergence rates, which are hereby obtained by using the discretization of a suitable linear elliptic operator as preconditioning matrix. In the present paper we are concerned with a nonlinear analogue of this idea, and propose that the Sobolev gradient approach, coupled with the preconditioning operator idea, provides an efficient organized framework of iterative methods for nonlinear elliptic problems. The Sobolev gradient approach has been developed in a series of publications [21]– [24], for a summary see the monograph [25]. (Related ideas are used in the so-called H-methods, see e.g. in [5, 27].) In the Sobolev gradient approach the iteration is constructed as a gradient (steepest descent) method for a suitable functional. The main principle of Sobolev gradients is that preconditioning can be obtained via a change of inner product to determine the gradient of the functional. In particular, a sometimes dramatic improvement can be achieved by using the Sobolev inner product instead of the original L one. This change appears in the iterative sequence as a preconditioning operator (or rather its discrete version when the algebraic system arising from FEM or FDM discretization of the PDE is solved). This preconditioning operator is the minus Laplacian or Su ≡ −∆u+cu when the standard Sobolev inner product is used, and, more generally, a suitable general elliptic operator when a weighted Sobolev inner product is applied (see e.g. [18]). The scope of Sobolev gradients includes least-square functionals for general operators, but in this paper we consider elliptic potential operators and minimize the corresponding potential. In this context the above-mentioned operator in the iterative sequence leads to the concept of preconditioning operators from the monograph [7]. Here a general iterative solution method for a nonlinear elliptic BVP F (u) = b is given by the projection of a sequence