# A variational parameters-to-estimate-free nonlinear solver

## Abstract

The paper introduces and analyzes an extension of the simple steepest descent variational iterative method for solving nonlinear equations defined from non-differentiable nonlinear mappings. This method is originally proposed for solving a class of nonlinear equations that arise in inner-outer iterative methods for solving linear equations with matrices of a two-by-two block form, see Axelsson and Vassilevski [2]. With minor modifications it turns out to be applicable to a more general class of nonlinear equations defined from non-differentiable mappings that are however sufficiently close to differentiable mappings in a neighborhood of the solution. An extension of the preconditioned steepest descent variational (PSD) method for solving semi-linear elliptic PDEs is illustrated with numerical experiments for a class of finite element discretization techniques based on two subspaces. The convergence of the PSD versus versions of Newton's method is compared and discussed.

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