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.
This research has been supported in part by the Bulgarian Ministry of Education, Science and Technology under Grants I-504/1995 and MM-415/1994.
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© 1997 Springer-Verlag Berlin Heidelberg
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Petrova, S.I., Vassilevski, P.S. (1997). A variational parameters-to-estimate-free nonlinear solver. In: Vulkov, L., Waśniewski, J., Yalamov, P. (eds) Numerical Analysis and Its Applications. WNAA 1996. Lecture Notes in Computer Science, vol 1196. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62598-4_119
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DOI: https://doi.org/10.1007/3-540-62598-4_119
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