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Incremental unknowns for solving nonlinear eigenvalue problems: New multiresolution methods. (English) Zbl 0828.65124
Nonlinear eigenvalue problems of type \(-\Delta u = \gamma u - \nu |u|u\) in \(\Omega = [0,1]^2\), \(u |_{\partial \Omega} = 0\) are considered. The authors discuss numerical schemes for obtaining \(u\) when the parameters \(\gamma\) and \(\nu\) are given. If difference systems \(AX = F(X)\) are written as \(X = T(X)\) then the basic iterations \(X^{K + 1} = X^K - \alpha (X^K - 2T (X^K) + T^2 (X^K))\) can be applied.
The authors suggest to use special multilevel difference schemes based on the use of hierarchical bases (the incremental unknowns method). This leads to matrices \(A\) with good condition numbers and relatively efficient numerical algorithms.
Reviewer’s remark: The above basic iterations assume actually that systems with the matrix \(A\) are solved exactly. Other iterations without this assumption can be found in the book of E. G. D’yakonov [“Optimization in solving elliptic problems”, CRC Press, Boca Raton, 1995].

MSC:
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
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