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Convergence Analysis of a Minimax Method for Finding Multiple Solutions of Semilinear Elliptic Equation: Part I—On Polyhedral Domain

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Abstract

In 2001 and 2002, Li–Zhou designed minimax methods for finding saddle critical points in Hilbert space. These methods were applied to numerically solve semilinear elliptic equation for multiple solutions. Subsequence and sequence convergence results for the methods in functional analysis were established in 2002. But, since these convergence results do not consider discretization, they are not convergence results in numerical analysis. In this paper, we point out what approximation problem is, when Li–Zhou’s methods are used to solve semilinear elliptic equation and the finite element method is used in discretization. Global sequence convergence result on computation of the approximation problem is verified. Finally, as diameter of element domain goes to zero, convergence of solutions of the approximation problem to solutions of semilinear elliptic equation is proved.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Xudong Yao

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  1. Xudong Yao
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Yao, X. Convergence Analysis of a Minimax Method for Finding Multiple Solutions of Semilinear Elliptic Equation: Part I—On Polyhedral Domain. J Sci Comput 62, 652–673 (2015). https://doi.org/10.1007/s10915-014-9871-8

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  • DOI: https://doi.org/10.1007/s10915-014-9871-8

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