Estimation of interpolation error constants for the P0 and P1 triangular finite elements

We give some fundamental results on the error constants for the piecewise constant interpolation function and the piecewise linear one over triangles. For the piecewise linear one, we mainly analyze the conforming case, but some results are also given for the non-conforming case. We obtain explicit relations for the dependence of such error constants on the geometric parameters of triangles. In particular, we explicitly determine the Babuška–Aziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element. The equation for determination is the transcendental equation λ+tanλ=0, so that the solution can be numerically obtained with desired accuracy and verification. Such highly accurate approximate values for the constant as well as estimates for other constants can be widely used for a priori and a posteriori error estimations in adaptive computation and numerical verification of finite element solutions.