An Interpolation Error Estimate on Anisotropic Meshes in Rn and Optimal Metrics for Mesh Refinement

In this paper, we extend the work in [W. Cao, Math. Comp., to appear] to functions of $n$ dimensions. We measure the anisotropic behavior of higher-order derivative tensors by the “largest” (in certain sense) ellipse/ellipsoid contained in the level curve/surface of the polynomial for directional derivatives. Given the anisotropic measure for the interpolated functions, we derive an error estimate for piecewise polynomial interpolations on meshes that are quasi-uniform under a given metric. By using the inertia properties for matrix eigenvalues [R. C. Thompson, J. Math. Anal. Appl., 58 (1977), pp. 572-577] and Holder's inequality, we can identify the optimal mesh metrics leading to the smallest error bound in various norms. Furthermore, we develop a dimensional reduction method to find the anisotropic measure approximately. We present two numerical examples for linear and quadratic interpolation on various anisotropic meshes generated with the optimal mesh metrics developed in this paper. Numerical results show that the smallest interpolation error is attained exactly on meshes optimal for the corresponding error norm as predicted.

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