Tau-lines: A new hybrid approach to the numerical treatment of crack problems based on the Tau method.

*(English)*Zbl 0576.73095A new hybrid computaional technique based on Ortiz’ recursive formulation of the Tau method is introduced in this paper and applied to some model singular boundary value problems which are relevant to fracture mechanics (modes I and III). This technique, which we call Tau-lines, combines the method of lines with the Tau method. The former is used in the construction of a system of coupled ordinary differential equations which is the discretized model of a given partial differential equation; the latter is used to find an accurate approximation of the solution of such a system which involves no further discretization. Recent theoretical results on the Tau method show that its error is optimal in the sense that, for a given degree n, it has the same order of error as the best uniform approximation of the exact solution by algebraic polynomials of degree n. The present work may be considered as an encouraging first step towards the development of the Tau-lines approach into a useful and efficient computational tool for the numerical treatment of problems in fracture mechanics.

##### MSC:

74R05 | Brittle damage |

65N40 | Method of lines for boundary value problems involving PDEs |

65J99 | Numerical analysis in abstract spaces |

65N99 | Numerical methods for partial differential equations, boundary value problems |

74S99 | Numerical and other methods in solid mechanics |

##### Keywords:

hybrid computational technique; Ortiz’ recursive formulation of the Tau method; singular boundary value problems; modes I and III; Tau-lines; method of lines; system of coupled ordinary differential equations; discretized model of a given partial differential equation; accurate approximation
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\textit{A. E. M. El Misiery} and \textit{E. L. Ortiz}, Comput. Methods Appl. Mech. Eng. 56, 265--282 (1986; Zbl 0576.73095)

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