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 H1-seminorm and L2-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 H1-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 √ |λ1/λ2| or less, where λ1 and λ2 are the eigenvalues of the Hessian matrix of the function. With these formulas we identify the optimal triangles, which produce the smallest H1-seminorm of the interpolation error, to be the acute isosceles aligned with the solution and of an aspect ratio about 0.8|1 λ2 |. The L2norm 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 L2-norm are those aligned with the solution and of an aspect ratio √ |λ1/λ2|. These formulas can be used to formulate more accurate mesh quality measures and to derive tighter error bounds for interpolations.