A nonlinear adaptive multiresolution method in finite differences with incremental unknowns. (English) Zbl 0836.65114

A new method for the calculation of unstable solutions of nonlinear eigenfunction problems is proposed. The method is derived from the classical Marder-Weitzner scheme (MW) which is treated as a nonlinear Richardson method. The method is generalized with the utilization of the incremental unknowns inducing the minimizing relaxation parameter in the embedded hierarchical subspaces. In this way the author obtains generalisations of the MW and linear Richardson algorithms. The numerical illustrations allowing comparisons between different versions of the MW method (for nonlinear eigenfunction problems) point out a better rate of convergence of the new algorithms.
Reviewer: P.Burda (Praha)


65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
Full Text: DOI EuDML


[1] C. BOLLEY, 1978, Multiple Solutions of a Bifurcation Problem, in Bifurcation and Nonlinear Eigenvalue Problems, ed. C. Bardos, Proceedings Univ. Paris XIII Villetaneuse, Springer Verlag, n^\circ 782, 42-53. Zbl0435.35012 MR572250 · Zbl 0435.35012
[2] J.-P. CHEHAB, R. TEMAM, Incremental Unknowns for Solving Nonlinear Eigenvalue Problems. New Multiresolution Methods, Numerical Methods for PDE’s, 11, 199-228 (1995). Zbl0828.65124 MR1325394 · Zbl 0828.65124 · doi:10.1002/num.1690110304
[3] M. CHEN, R. TEMAM, 1991, Incremental Unknowns for Solving Partial Differential Equations, Numerische Matematik, 59, 255-271. Zbl0712.65103 MR1106383 · Zbl 0712.65103 · doi:10.1007/BF01385779
[4] M. CHEN, R. TEMAM, 1993, Incremental Unknowns in Finite Differences : Condition Number of the Matrix, SIAM J. of Matrix Analysis and Applications (SIMAX), 14, n^\circ 2, 432-455. Zbl0773.65080 MR1211799 · Zbl 0773.65080 · doi:10.1137/0614031
[5] G. H. GOLUB, G. A. MEURANT, 1983, Résolution numérique des grands systèmes linéaires, Ecole d’été d’Analyse Numérique CEA-EDF-INRIA, Eyrolles. Zbl0646.65022 MR756627 · Zbl 0646.65022
[6] D. HENRY, 1981, Geometric Theory of Semilinear Parabolic Equations, Springer Verlag, n^\circ 840. Zbl0456.35001 MR610244 · Zbl 0456.35001
[7] H. MARDER, B. WEITZNER, 1970, A Bifurcation Problem in E-layer Equilibria, Plasma Physics, 12, 435-445. Zbl0195.29002 · Zbl 0195.29002
[8] M. MARION, R. TEMAM, 1989, Nonlinear Galerkin Methods, SIAM Journal of Numerical Analysis, 26, 1139-1157. Zbl0683.65083 MR1014878 · Zbl 0683.65083 · doi:10.1137/0726063
[9] M. MARION, R. TEMAM, 1990, Nonlinear Galerkin Methods ; The Finite elements case, Numerische Matematik, 57, 205-226. Zbl0702.65081 MR1057121 · Zbl 0702.65081 · doi:10.1007/BF01386407
[10] M. SERMANGE, 1979, Une méthode numérique en bifurcation. Application à un problème à frontière libre de la physique des plasmas, Applied Mathematics and Optimization, 127-151. Zbl0393.65026 MR533616 · Zbl 0393.65026 · doi:10.1007/BF01442550
[11] R. TEMAM, 1990, Inertial Manifolds and Multigrid Methods, SIAM J. Math. Anal, 21, 154-178. Zbl0715.35039 MR1032732 · Zbl 0715.35039 · doi:10.1137/0521009
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