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Finite elements of arbitrary order and quasiinterpolation for data in Riemannian manifolds

by P. Grohs

(Report number 2011-56)

Abstract
We consider quasiinterpolation operators for functions assuming their values in a Riemannian manifold. We construct such operators from corresponding linear quasiinterpolation operators by replacing affine averages with the Riemannian center of mass. As a main result we show that the approximation rate of such a nonlinear operator is the same as for the linear operator it has been derived from. In order to formulate this result in an intrinsic way we use the Sasaki metric to compare the derivatives of the function to be approximated with the derivatives of the nonlinear approximant. Numerical experiments confirm our theoretical findings.

Keywords:

@Techreport{G11_128,
  author = {P. Grohs},
  title = {Finite elements of arbitrary order and quasiinterpolation for data in Riemannian manifolds},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-56},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-56.pdf },
  year = {2011}
}

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