##
**A finite element collocation method for singular integral equations.**
*(English)*
Zbl 0543.65089

The approximate solution of integral equations on closed curves with splines as trial functions and collocation at the break points for odd degree (and at the midpoints of the partitioning intervals for even degree) splines corresponds to one of the most extensively used boundary element methods in engineering computations for two-dimensional boundary value problems. However, since V. V. Ivanov showed in his book [The theory of approximate methods and their application to the numerical solution of singular integral equations (1976; Zbl 0346.65065), pp. 200 ff.] that for the Hilbert transform S this spline collocation defines an unstable divergent method, the convergence of spline collocation was only justified for Fredholm integral equations of the second kind with smooth kernel.

Here the authors present the first rigorous error analysis for singular integral equations \(a(t)x(t)+b(t) S x(t)=y(t)\) on the unit circle with Cauchy kernel, piecewise linear trial functions and collocation at the break points. They show that strong ellipticity of the equation is sufficient and necessary for convergence and show optimal order convergence in Hölder spaces.

Strong ellipticity means that there exist a regular function \(\alpha\) (t) and a positive constant \(\theta\) such that \(Re \theta(t)(a(t)\pm b(t))\geq \alpha>0\) holds for all \(| t| =1\), which is equivalent to \(a(t)+\lambda b(t)\neq 0\) for all \(| t| =1\) and \(\lambda \in [- 1,1].\)

The results are obtained first for constant coefficients a,b via the explicitly known spectrum of the circulant matrices defining the discrete approximate equations. Then they are extended to the case of continuous a(t), b(t) with perturbation arguments of the functional analysis of discretization methods.

The restriction to strongly elliptic equations reveals a big difference between spline collocation and collocation methods for the global trigonometric polynomials [cf. the first author and B. Silbermann, Projektionsverfahren und die näherungsweise Lösung singulärer Gleichungen (1977; Zbl 0364.65044)] since the latter converge for the much larger class of equations with index zero. This paper has stimulated a whole series of new investigations and new results on one-dimensional spline collocation, see e.g. the reviewers survey article [On the spline approximation of singular integral equations and one-dimensional pseudo- differential equations on closed curves. In: A. Gerasoulis, R. Vichnevetsky (eds.), Numerical Solution of Singular Integral Equations. IMACS Publ., Rutgers Univ. New Brunswick U.S.A., 113-119 (1984)] and the recent article by the second author [On spline collocation methods for boundary integral equations in the plane. Math. Methods Appl. Sci. (to appear)].

Here the authors present the first rigorous error analysis for singular integral equations \(a(t)x(t)+b(t) S x(t)=y(t)\) on the unit circle with Cauchy kernel, piecewise linear trial functions and collocation at the break points. They show that strong ellipticity of the equation is sufficient and necessary for convergence and show optimal order convergence in Hölder spaces.

Strong ellipticity means that there exist a regular function \(\alpha\) (t) and a positive constant \(\theta\) such that \(Re \theta(t)(a(t)\pm b(t))\geq \alpha>0\) holds for all \(| t| =1\), which is equivalent to \(a(t)+\lambda b(t)\neq 0\) for all \(| t| =1\) and \(\lambda \in [- 1,1].\)

The results are obtained first for constant coefficients a,b via the explicitly known spectrum of the circulant matrices defining the discrete approximate equations. Then they are extended to the case of continuous a(t), b(t) with perturbation arguments of the functional analysis of discretization methods.

The restriction to strongly elliptic equations reveals a big difference between spline collocation and collocation methods for the global trigonometric polynomials [cf. the first author and B. Silbermann, Projektionsverfahren und die näherungsweise Lösung singulärer Gleichungen (1977; Zbl 0364.65044)] since the latter converge for the much larger class of equations with index zero. This paper has stimulated a whole series of new investigations and new results on one-dimensional spline collocation, see e.g. the reviewers survey article [On the spline approximation of singular integral equations and one-dimensional pseudo- differential equations on closed curves. In: A. Gerasoulis, R. Vichnevetsky (eds.), Numerical Solution of Singular Integral Equations. IMACS Publ., Rutgers Univ. New Brunswick U.S.A., 113-119 (1984)] and the recent article by the second author [On spline collocation methods for boundary integral equations in the plane. Math. Methods Appl. Sci. (to appear)].

Reviewer: W.Wendland

### MSC:

65R20 | Numerical methods for integral equations |

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

45E05 | Integral equations with kernels of Cauchy type |

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

35C15 | Integral representations of solutions to PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

### Keywords:

spline collocation; error analysis; piecewise linear trial functions; strong; optimal order convergence; Hölder spaces
PDF
BibTeX
XML
Cite

\textit{S. Prößdorf} and \textit{G. Schmidt}, Math. Nachr. 100, 33--60 (1981; Zbl 0543.65089)

Full Text:
DOI

### References:

[1] | A survey of numerical methods for the solution of Fredholm integral equations of the second kind, SIAM, Philadelphia 1976. |

[2] | , Higher transeendental functions, vol. 1, Mc Graw-Hill, New York 1973. |

[3] | Carlson, Pacific. J. of math. 15 pp 1119– (1965) · Zbl 0142.27001 |

[4] | [Russian text Ignored] 3 pp 51– (1965) |

[5] | [Russian text Ignored] 1977. |

[6] | Hildebrandt, Comm. pure and appl. math. pp 369– (1964) |

[7] | Hsiao, Computing 25 pp 89– (1980) |

[8] | The theory of approximate methods and their application to the numerical solution of singular integral equations, Noordhoff, Leyden 1976. |

[9] | Kahan, Proc. AMS 39 pp 235– (1973) |

[10] | , Funktionalanalysis in normierten Räumen, Akad.-Verl., Berlin 1978. |

[11] | Perturbations theory for linear operators, Springer, Berlin 1966. |

[12] | M. A. [Russian text Ignored] 118, 3–56 (1932). |

[13] | , A survey of matrix theory and matrix inequalities, Allyn and Bacon, Boston 1964. |

[14] | , Singuläre Integraloperatoren, Akad.-Verl., Berlin 1980. |

[15] | Some classes of singular equations, North Holand, Amsterdam 1978. |

[16] | , Projektionsverfahren und die näherungsweise Lösung singulärer Gleichungen, Teubner, Leipzig 1977. · Zbl 0364.65044 |

[17] | [Russian text Ignored] 1 pp 37– (1956) |

[18] | , Remarks to GALERKIN and least squares methods with finite elements for general elliptic problems, Lecture Notes Math. 564, 461–471, Springer, Berlin 1976. |

[19] | Funktionalanalysis der Diskretisierungsmethoden, Teubner, Leipzig 1976. |

[20] | Wendland, Intern. Series Num. Math. 53 pp 244– (1980) |

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.