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Collocation with WEB–Splines

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Abstract

We describe a collocation method with weighted extended B–splines (WEB–splines) for arbitrary bounded multidimensional domains, considering Poisson’s equation as a typical model problem. By slightly modifying the B–spline classification for the WEB–basis, the centers of the supports of inner B–splines can be used as collocation points. This resolves the mismatch between the number of basis functions and interpolation conditions, already present in classical univariate schemes, in a simple fashion. Collocation with WEB–splines is particularly easy to implement when the domain boundary can be represented as zero set of a weight function; sample programs are provided on the website http://www.web-spline.de. In contrast to standard finite element methods, no mesh generation and numerical integration is required, regardless of the geometric shape of the domain. As a consequence, the system equations can be compiled very efficiently. Moreover, numerical tests confirm that increasing the B–spline degree yields highly accurate approximations already on relatively coarse grids. Compared with Ritz-Galerkin methods, the observed convergence rates are decreased by 1 or 2 when using splines of odd or even order, respectively. This drawback, however, is outweighed by a substantially smaller bandwidth of collocation matrices.

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Authors and Affiliations

  1. IMNG, Fachbereich Mathematik, Universität Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Christian Apprich, Klaus Höllig & Jörg Hörner

  2. AG Geometrie & Approximation, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Ulrich Reif

Authors
  1. Christian Apprich
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  2. Klaus Höllig
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  3. Jörg Hörner
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  4. Ulrich Reif
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Correspondence to Ulrich Reif.

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Communicated by: T. Lyche

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Apprich, C., Höllig, K., Hörner, J. et al. Collocation with WEB–Splines. Adv Comput Math 42, 823–842 (2016). https://doi.org/10.1007/s10444-015-9444-x

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