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- W1992595544 abstract "The main purpose of this paper is to present a closer look at how the $H^1$‐ and $L^2$‐errors for quadratic interpolation on a triangle are determined by the triangle geometry and the anisotropic behavior of the third order derivatives of interpolated functions. We characterize quantitatively the anisotropic behavior of a third order derivative tensor by its orientation and anisotropic ratio. Both exact error formulas and numerical experiments are presented for model problems of interpolating a cubic function u at the vertices and the midpoints of three sides of a triangle. Based on the study on model problems, we conclude that when an element is aligned with the orientation of $nabla^3 u$, the aspect ratio leading to nearly the smallest $H^1$‐ and $L^2$‐norms of the interpolation error is approximately equal to the anisotropic ratio of $nabla^3 u$. With this alignment and aspect ratio taken, the $H^1$‐seminorm of the error is proportional to the reciprocal of the anisotropic ratio of $nabla^3 u$, the $L^2$‐norm of the error is proportional to the $-frac 32$th power of the anisotropic ratio, and both of them are insensitive to the internal angles of the element." @default.
- W1992595544 created "2016-06-24" @default.
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- W1992595544 date "2007-01-01" @default.
- W1992595544 modified "2023-09-26" @default.
- W1992595544 title "Anisotropic Measures of Third Order Derivatives and the Quadratic Interpolation Error on Triangular Elements" @default.
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- W1992595544 doi "https://doi.org/10.1137/050634700" @default.
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