An interpolation error estimate in \(\mathcal{R}^2\) based on the anisotropic measures of higher order derivatives. (English) Zbl 1149.65010

The main result of the paper establishes an anisotropic error estimate for interpolation of a function \(u\) by piecewise polynomials of degree \(k\geq 1\) over triangulations that are quasi-uniform under a given Riemannian metric. Based on the estimate, an optimal metric corresponding to the smallest error bound is identified. This extends the optimal interpolation error estimates already obtained in the literature for linear elements to higher order bivariate elements.
The paper relates the interpolation error to the geometric features of an anisotropic triangular element in terms of the magnitude, orientation, and anisotropic ratio of the higher order derivative \(\nabla^{k+1} u\). These three quantities introduced by the author are invariant under translation and rotation of the \(xy\)-coordinates. A numerical example which supports the predicted optimal metric is presented, and the importance of the main result for anisotropic mesh generation and refinement is discussed.


65D05 Numerical interpolation
41A05 Interpolation in approximation theory
41A10 Approximation by polynomials
Full Text: DOI


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