New solutions of the Gelfand problem. (English) Zbl 1003.65131
A computer-assisted method for showing the existence of some non-symmetric solutions to the Bratu-Gelfand problem on non-convex domains in the plane is given based on an enclosure theorem for weak solutions. The numerical procedure involves a finite element discretization and interval methods for solving it.
Reviewer: Eugene L.Allgower (Fort Collins)
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
| 65G20 | Algorithms with automatic result verification |
| 65G40 | General methods in interval analysis |
Keywords:
Bratu-Gelfand problem; solution enclosures; computer-assisted method; existence; non-symmetric solutions; weak solutions; finite element; interval methodsReferences:
| [1] | Gelfand, I. M., Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl. Ser., 2 29, 295-381 (1963) · Zbl 0127.04901 |
| [2] | Amann, H., Existence of multiple solutions for nonlinear boundary value problems, Indiana Univ. Math. J., 21, 925-935 (1972) · Zbl 0222.35023 |
| [3] | Ambrosetti, A.; Prodi, G., On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Math. Pura Appl., 93, 231-247 (1973) · Zbl 0288.35020 |
| [4] | Bandle, C., Existence theorems, qualitative results and a priori bounds for a class of nonlinear Dirichlet problems, Arch. Rational Mech. Anal., 58, 219-238 (1975) · Zbl 0335.35046 |
| [5] | Joseph, D. D.; Lundgren, T. S., Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49, 241-269 (1973) · Zbl 0266.34021 |
| [6] | Keller, H. B.; Keener, J. P., Positive solutions of convex nonlinear eigenvalue problems, J. Differential Equations, 16, 103-125 (1974) · Zbl 0287.35074 |
| [7] | Kazdar, J. L.; Warner, F., Remarks on quasilinear elliptic equations, Comm. Pure Appl. Math., 28, 567-597 (1975) · Zbl 0325.35038 |
| [8] | Crandall, M. G.; Rabinovitz, P. H., Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal., 58, 207-218 (1975) · Zbl 0309.35057 |
| [9] | Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18, 620-709 (1976) · Zbl 0345.47044 |
| [10] | Brezis, H.; Turner, R. E.L., On a class of superlinear elliptic problems, Comm. Partial Differential Equations, 2, 601-614 (1977) · Zbl 0358.35032 |
| [11] | Gidas, B.; Ni, W. M.; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68, 209-243 (1979) · Zbl 0425.35020 |
| [12] | Mignot, F.; Puel, J.-P., Sur une classe de problemes non lineaires aves une linearite positive, croissante, convexe, Comm. Partial Differential Equations, 5, 791-836 (1980) · Zbl 0456.35034 |
| [13] | Lazer, A. C.; McKenna, P. J., On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl., 84, 282-294 (1981) · Zbl 0496.35039 |
| [14] | Lions, P. L., On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24, 441-467 (1982) · Zbl 0511.35033 |
| [15] | Hofer, H., Variational and topological methods in partially ordered Hilbert spaces, Math. Ann., 261, 493-514 (1982) · Zbl 0488.47034 |
| [16] | Breuer, B.; Plum, M.; McKenna, P. J., Inclusions and existence proofs for solutions of a nonlinear boundary value problem by spectral numerical methods, Comput. Suppl., 15, 61-77 (2001) · Zbl 0994.65129 |
| [17] | Plum, M., Enclosures for weak solutions of nonlinear elliptic boundary value problems, (Agarwal, R. P., Inequalities and Applications. Inequalities and Applications, World Scientific Series in Applicable Analysis, 3 (1994)), 505-521 · Zbl 0900.35146 |
| [18] | Plum, M., Computer-assisted enclosure methods for elliptic differential equations, Linear Algebra Appl., 324, 147-187 (2001) · Zbl 0973.65100 |
| [19] | Nakao, M. T., Solving nonlinear elliptic problems with result verification using an \(H^{−1}\) type residual iteration, Comput. Suppl., 9, 161-173 (1993) · Zbl 0802.65111 |
| [20] | Plum, M., Enclosures for solutions of parameter-dependent nonlinear elliptic boundary value problems: Theory and implementation on a parallel computer, Interval Comput., 3, 106-121 (1994) · Zbl 0833.65134 |
| [21] | Wieners, C., A parallel Newton multigrid method for high order finite elements and its application to numerical existence proofs for elliptic boundary value equations, Z. Angew. Math. Mech., 76, 171-176 (1996) |
| [22] | Plum, M., Enclosures for two-point boundary value problems near bifurcation points, (Alefeld, G.; Frommer, A.; Lang, B., Scientific Computation and Validated Numerics (1996), Akademie Verlag), 265-279 · Zbl 0849.65060 |
| [23] | Moser, J., A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20, 1077-1092 (1971) · Zbl 0203.43701 |
| [24] | Ladyzhenskaya, O.; Uraltseva, N., Linear and Quasilinear Elliptic Equations (1968), Academic Press: Academic Press New York · Zbl 0164.13002 |
| [25] | Bastian, P.; Birken, K.; Johannsen, K.; Lang, S.; Neuß, N.; Rentz-Reichert, H.; Wieners, C., UG—a flexible software toolbox for solving partial differential equations, Comput. Visualization Sci., 1, 27-40 (1997) · Zbl 0970.65129 |
| [26] | Klatte, R.; Kulisch, U.; Lawo, C.; Rausch, M.; Wiethoff, A., C-XSC—A C++ Class Library for Extended Scientific Computing (1993), Springer · Zbl 0814.68035 |
| [27] | Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer · Zbl 0788.73002 |
| [28] | Behnke, H.; Mertins, U.; Plum, M.; Wieners, C., Eigenvalue inclusions via domain decomposition, Proc. Roy. Soc. London Ser. A, 456, 2717-2730 (2000) · Zbl 0993.65120 |
| [29] | Behnke, H.; Goerisch, F., Inclusions for eigenvalues of selfadjoint problems, (Herzberger, J., Topics in Validated Computations. Topics in Validated Computations, Series Studies in Computational Mathematics (1994), North-Holland), 277-322 · Zbl 0838.65060 |
| [30] | Plum, M., Guaranteed numerical bounds for eigenvalues, (Hinton, D.; Schaefer, W., Spectral Theory and Computational Methods of Sturm-Liouville Problems. Spectral Theory and Computational Methods of Sturm-Liouville Problems, Lecture Notes in Pure and Applied Mathematics, 191 (1997)), 313-332 · Zbl 0885.47006 |
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.
