Parallel finite element splitting-up method for parabolic problems.

*(English)*Zbl 0747.65084The paper treats a method for solving parabolic problems using a splitting-up principle, i.e. the 2D-problem is first discretized in time, afterwards a 1D-subproblem, fixing \(y\), is solved in \(x\)-direction. Moving back the information at the time level \(t_{i+1}\) to the time level \(t_ i\), another 1D-problem is solved in \(y\)-direction. The solution of the 1D-problem is accomplished by applying cubic \(B\)- splines.

With the help of semigroup theory the convergence of the method is proved. A few examples conclude the careful investigation, showing in addition that the use of parallel processors is possible.

With the help of semigroup theory the convergence of the method is proved. A few examples conclude the careful investigation, showing in addition that the use of parallel processors is possible.

Reviewer: N.Herrmann (Hannover)

##### MSC:

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65Y05 | Parallel numerical computation |

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

35K15 | Initial value problems for second-order parabolic equations |

##### Keywords:

parabolic problems; splitting-up method; finite element method; parallel computation; cubic \(B\)-splines; semigroup theory; convergence
PDF
BibTeX
XML
Cite

\textit{X. Tai} and \textit{P. Neittaanmäki}, Numer. Methods Partial Differ. Equations 7, No. 3, 209--225 (1991; Zbl 0747.65084)

Full Text:
DOI

**OpenURL**

##### References:

[1] | ”Splitting methods for time dependent partial differential equations,” in The State of the Art in Numerical Analysis, Academic, London, 1977, pp. 757-796. |

[2] | ”Method of numerical mathematics,” Springer-Verlag, Berlin, 1975. |

[3] | The Method of Fractional Steps, Springer-Verlag, Berlin, 1971. |

[4] | and , ”Alternating direction Galerkin method on rectangles,” in Numerical Solution of Partial Differential Equations II, Ed., Academic, London, 1971, pp. 133-214. |

[5] | Marchuk, J. Comput. Phys. 52 pp 237– (1983) |

[6] | Bramble, SIAM J. Numer. Anal. 26 pp 904– (1989) |

[7] | and , ”A FE-splitting-up method and its application to distributed parameter identification of parabolic type,” (unpublished). |

[8] | Semigroup of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, Berlin, 1983. · Zbl 0516.47023 |

[9] | The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. |

[10] | and , ”Finite element approximations of variational problems with applications,” Pitman monographs and surveys in pure and applied mathematics, 50, Longman, 1990. |

[11] | , and , ”Linear and quasilinear equation of parabolic type,” Amer. Math. Soc., Providence, RI, 1968. |

[12] | Spline and Variational Method, Wiley, Chichester, UK, 1975. |

[13] | Spline Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1973. · Zbl 0333.41009 |

[14] | Gourlay, BIT 7 pp 31– (1967) |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.