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Sard kernel theorems on triangular domains with application to finite element error bounds

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Summary

Error bounds for interpolation remainders on triangles are derived by means of extensions of the Sard Kernel Theorems. These bounds are applied to the Galerkin method for elliptic boundary value problems. Certain kernels are shown to be identically zero under hypotheses which are, for example, fulfilled by tensor product interpolants on rectangles. This removes certain restrictions on how the sides of the triangles and/or rectangles tend to zero. Explicit error bounds are computed for piecewise linear interpolation over a triangulation and applied to a model problem.

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Authors and Affiliations

  1. Mathematics Department, University of Utah, 84112, Salt Lake City, Utah, USA

    Robert E. Barnhill

  2. Mathematics Department, Brunel University, Uxbridge, Middlesex, England

    John A. Gregory

Authors
  1. Robert E. Barnhill
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  2. John A. Gregory
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Additional information

The research of R. E. Barnhill was supported by The National Science Foundation with Grant GP 20293 to the University of Utah, the Science Research Council with Grants B/SR/9652 at Brunel University and B/RG/61876 at Dundee University, a N.A.T.O. Senior fellowship in Science, and The University of Utah Research Committee.

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Barnhill, R.E., Gregory, J.A. Sard kernel theorems on triangular domains with application to finite element error bounds. Numer. Math. 25, 215–229 (1975). https://doi.org/10.1007/BF01399411

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