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Numerische Mathematik

, Volume 62, Issue 1, pp 511–538 | Cite as

Spline collocation for strongly elliptic equations on the torus

  • Martin Costabel
  • William McLean
Article

Summary

We prove convergence and error estimates in Sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus. Examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus-shaped domains in three or more dimensions. For odd-degree splines, we prove convergence of nodal collocation for any strongly elliptic operator. For even-degree splines and midpoint collocation, we find an additional condition for the convergence which is satisfied for the classical boundary integral operators. Our analysis is a generalization to higher dimensions of the corresponding analysis of Arnold and Wendland [4].

Mathematics Subject Classification (1991)

65R20 (35S15 41A15 41A63 65N35 

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Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Martin Costabel
    • 1
  • William McLean
    • 2
  1. 1.Département de MathématiquesUniversité Bordeaux 1Talence CédexFrance
  2. 2.School of MathematicsUniversity of New South WalesSydneyAustralia

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