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**Anisotropic measures of third order derivatives and the quadratic interpolation error on triangular elements.**
*(English)*
Zbl 1136.65100

Summary: 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 behaviour of the third order derivatives of interpolated functions. We characterize quantitatively the anisotropic behaviour 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.

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.

### MSC:

65N15 | Error bounds for boundary value problems involving PDEs |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65D05 | Numerical interpolation |

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |

35J25 | Boundary value problems for second-order elliptic equations |