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Preconditioning operators and Sobolev gradients for nonlinear elliptic problems. (English) Zbl 1118.65122

The main goal of the authors is to develop a general framework for the preconditioning of nonlinear elliptic boundary value problems using fixed preconditioning operators in each iteration. In the case of nonlinear problems, the preconditioners may vary in course of the iteration, in which case the study of the iteration can be best based on a quasi-Newton method framework.
The authors focus on simple iterations, i.e., when fixed preconditioning operators are used. The main advantages of fixed preconditioning operators are the lack of need to update and the use of fast solvers which are only available for certain particular types of elliptic operators. In the context of the Sobolev gradient technique, the approach of the paper can be interpreted as descent with respect to weighted Sobolev inner products.
In the first part of the paper, results on linear operators are reviewed to provide suitable background for the next sections. The second part contains main theorems devoted to the infinite-dimensional preconditioning of nonlinear operators. The last part is essential from a numerical point of view since it is devoted to preconditioning strategies for the discretized elliptic problems. The discrete Laplacian preconditioner, separable preconditioners, initial shape preconditioners and also preconditioners for fourth-order problems are delivered.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
65F35 Numerical computation of matrix norms, conditioning, scaling
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