×

Superconvergence of FEMs and numerical continuation for parameter-dependent problems with folds. (English) Zbl 1147.65324

Summary: We study finite element approximations for positive solutions of semilinear elliptic eigenvalue problems with folds, and exploit the superconvergence of finite element methods (FEM). In order to apply the superconvergence of FEM for Poisson’s equation to parameter-dependent problems with folds, this paper provides the framework of analysis, accompanied with the proof of the strong monotonicity of the nonlinear form. It is worthy to point out that the superconvergence of the nonlinear problem in this paper is different from that of Chen and Huang. A continuation algorithm is described to trace solution curves of semilinear elliptic eigenvalue problems, where the Adini elements are exploited to discretize the PDEs. Numerical results on some sample test problems with folds and bifurcations are reported.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Software:

PITCON
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1137/1.9780898719154 · Zbl 1036.65047 · doi:10.1137/1.9780898719154
[2] Atkinson K. E., An Introduction to Numerical Analysis (1989) · Zbl 0718.65001
[3] DOI: 10.1137/0907036 · Zbl 0589.65074 · doi:10.1137/0907036
[4] DOI: 10.1007/BF01395985 · Zbl 0488.65021 · doi:10.1007/BF01395985
[5] DOI: 10.1007/BF01396184 · Zbl 0525.65036 · doi:10.1007/BF01396184
[6] DOI: 10.1007/BF01395805 · Zbl 0525.65037 · doi:10.1007/BF01395805
[7] G. Caloz and J. Rappaz, Handbook of Numerical Analysis V, eds. P. G. Ciarlet and J. L. Lions (Elsevier Science, BV, 1997) pp. 487–637.
[8] DOI: 10.1142/S0218127405013630 · Zbl 1092.65503 · doi:10.1142/S0218127405013630
[9] Chen C., High Accuracy Theory of Finite Element Methods (1995)
[10] DOI: 10.1002/nla.306 · Zbl 1071.65038 · doi:10.1002/nla.306
[11] DOI: 10.1142/S0218127405013319 · Zbl 1092.37528 · doi:10.1142/S0218127405013319
[12] DOI: 10.1137/030602447 · Zbl 1095.65100 · doi:10.1137/030602447
[13] P. G. Ciarlet, Finite Element Methods, eds. P. G. Ciarlet and J. L. Lions (North-Holland, Amsterdam, 1991) pp. 17–351. · Zbl 0850.65203
[14] DOI: 10.1137/0906055 · Zbl 0589.65075 · doi:10.1137/0906055
[15] DOI: 10.1137/1.9780898719543 · Zbl 0935.37054 · doi:10.1137/1.9780898719543
[16] Hiriart-Urruty J.-B., Convex Analysis and Minimization Algorithms I. Fundamentals (1991) · Zbl 0807.46041
[17] DOI: 10.1016/j.apnum.2003.10.009 · Zbl 1069.65119 · doi:10.1016/j.apnum.2003.10.009
[18] DOI: 10.1016/j.apnum.2005.06.003 · Zbl 1091.65111 · doi:10.1016/j.apnum.2005.06.003
[19] Keller H. B., Lectures on Numerical Methods in Bifurcation Problems (1987)
[20] DOI: 10.1007/978-1-4613-3338-8 · doi:10.1007/978-1-4613-3338-8
[21] Li Z. C., Computing 65 pp 27–
[22] DOI: 10.1016/S0168-9274(01)00135-0 · Zbl 1022.65121 · doi:10.1016/S0168-9274(01)00135-0
[23] Lin Q., The Construction and Analysis of High Efficient FEM (1996)
[24] Lin Q., Finite Element Methods, Accuracy and Improvements (2006)
[25] DOI: 10.1137/1024101 · Zbl 0511.35033 · doi:10.1137/1024101
[26] Rheinboldt W. C., Numerical Analysis of Parametrized Nonlinear Equations (1986) · Zbl 0582.65042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.