On the Error of Linear Interpolation and the Orientation, Aspect Ratio, and Internal Angles of a Triangle

In this paper, we attempt to reveal the precise relation between the error of linear interpolation on a general triangle and the geometric characters of the triangle. Taking the model problem of interpolating quadratic functions, we derive two exact formulas for the $H^1$-seminorm and $L^2$-norm of the interpolation error in terms of the area, aspect ratio, orientation, and internal angles of the triangle. These formulas indicate that (1) for highly anisotropic triangular meshes the $H^1$-seminorm of the interpolation error is almost a monotonically decreasing function of the angle between the orientations of the triangle and the function; (2) maximum angle condition is not essential if the mesh is aligned with the function and the aspect ratio is of magnitude $\sqrt{|\lambda_1/\lambda_2|}$ or less, where $\lambda_1$ and $\lambda_2$ are the eigenvalues of the Hessian matrix of the function. With these formulas we identify the optimal triangles, which produce the smallest $H^1$-seminorm of the interpolation error, to be the acute isosceles aligned with the solution and of an aspect ratio about $0.8 |\frac{\lambda_1}{\lambda_2}|$. The $L^2$-norm of the interpolation error depends on the orientation and the aspect ratio of the triangle, but not directly on its maximum or minimum angles. The optimal triangles for the $L^2$-norm are those aligned with the solution and of an aspect ratio $\sqrt{|\lambda_1/\lambda_2|}$. These formulas can be used to formulate more accurate mesh quality measures and to derive tighter error bounds for interpolations.