##
**High-order BEM formulations for strongly nonlinear problems governed by quite general nonlinear differential operators.**
*(English)*
Zbl 0882.65108

The author uses the technique of homotopy in order to obtain an integral formulation for some general nonlinear boundary value problem corresponding to partial differential equations of elliptic type. Therefore, the method refers to those nonlinear problems whose governing equations as well as boundary conditions do not contain any linear term at all.

Starting from such a very general boundary value problem, the author proposes a homotopy mapping and an iterative algorithm where, at each iterative step, a linear boundary value problem is solved by the classical boundary element method (BEM). This is called a general BEM and the author observes that it contains the traditional BEM; i.e. BEM applied to a differential operator which is a sum of a familiar linear operator and a nonlinear operator; consequently the linear operator could be inverted.

The idea seems to be quite valuable and the author has extended it to parabolic problems (Navier-Stokes equations). However, as he realizes, deeper mathematical insight is necessary to improve and justify this method. The present paper contains, as numerical examples, three nonlinear two-point boundary value problems for second-order differential operators.

Starting from such a very general boundary value problem, the author proposes a homotopy mapping and an iterative algorithm where, at each iterative step, a linear boundary value problem is solved by the classical boundary element method (BEM). This is called a general BEM and the author observes that it contains the traditional BEM; i.e. BEM applied to a differential operator which is a sum of a familiar linear operator and a nonlinear operator; consequently the linear operator could be inverted.

The idea seems to be quite valuable and the author has extended it to parabolic problems (Navier-Stokes equations). However, as he realizes, deeper mathematical insight is necessary to improve and justify this method. The present paper contains, as numerical examples, three nonlinear two-point boundary value problems for second-order differential operators.

Reviewer: C.I.Gheorghiu (Cluj-Napoca)

### MSC:

65N38 | Boundary element methods for boundary value problems involving PDEs |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

### Keywords:

strongly nonlinear problems; homotopy method; iterative algorithm; boundary element method; numerical examples
PDFBibTeX
XMLCite

\textit{S. Liao}, Int. J. Numer. Methods Fluids 23, No. 8, 739--751 (1996; Zbl 0882.65108)

Full Text:
DOI

### References:

[1] | The Boundary Element for Engineers, Pentech, London, 1980. |

[2] | and , Advances in Boundary Elements 1: Computations and Fundamentals, Computational Mechanics Publications, Southampton, 1989. |

[3] | Boundary Elements X, Vol. 1, Mathematical and Computational Aspects, Computational Mechanics Publications, Southampton, 1988. |

[4] | Boundary Methods–An Algebraic Theory, Pitman, Boston, MA, 1984. |

[5] | Boundary Value Problems of Linear Partial Differential Equations for Engineers and Scientists, World Scientific, Hong Kong, 1987. · doi:10.1142/0489 |

[6] | Liao, Int. j. numer. methods fluids 15 pp 595– (1992) · Zbl 0762.76063 · doi:10.1002/fld.1650150507 |

[7] | Liao, Int. j. numer. methods fluids 22 pp 1– (1996) · Zbl 0873.76052 · doi:10.1002/(SICI)1097-0363(19960115)22:1<1::AID-FLD314>3.0.CO;2-5 |

[8] | ’The quite general BEM for strongly non-linear problems’, in et al. (eds), Boundary Elements XVIII, Computational Mechanics Publications, Southampton, 1995, pp. 67-74. · Zbl 0839.65132 |

[9] | Rodriguez-Prada, Int. j. numer. methods fluids 10 (1990) · Zbl 0692.76025 · doi:10.1002/fld.1650100102 |

[10] | and , ’The generalized BEM for non-linear problems’, in (ed.), Boundary Elements X, Vol. 1, Mathematical and Computational Aspects, Computational Mechanics Publications, Southampton 1988, pp. 1-17. |

[11] | and , ’Integral equation analysis for geometrically non-linear-problems of elastic bodies’, in and (eds.), Theory and Applications of Boundary Element Methods, Pergamon, Oxford, 1987, pp. 251-260. |

[12] | ’A kind of linearly-invariance under homotopy and some simple applications of it in mechanics’, Report 520, Institute of Shipbuilding, University of Hamburg, 1992. |

[13] | Liao, Int. j. non-lin. mech. 30 pp 371– (1995) · Zbl 0837.76073 · doi:10.1016/0020-7462(94)00054-E |

[14] | Topology: A General Account of General Topology, Homotopy Types and the Fundamental Groupoid, Wiley, New York, 1988. · Zbl 0655.55001 |

[15] | ’New developments in the dual reciprocity method’, in , and (eds), Boundary Element XVII, Computational Mechanics Publications, Southampton, 1995, pp. 11-18. |

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.