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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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