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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)].
Reviewer: W.Wendland

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
Full Text: DOI
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