## Analysis and Estimation of Error Constants for {\large{\boldmath$P_0$}} and {\large{\boldmath$P_1$}} Interpolations over Triangular Finite Elements

J. Math. Sci. Univ. Tokyo
Vol. 17 (2010), No. 1, Page 27--78.

Liu, Xuefeng; Kikuchi, Fumio
Analysis and Estimation of Error Constants for {\large{\boldmath$P_0$}} and {\large{\boldmath$P_1$}} Interpolations over Triangular Finite Elements

Abstract:
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 the present results also appear to be available for the non-conforming case. We obtain explicit relations for the upper bounds of the constants, and analyze dependence of such constants on the geometric parameters of triangles. In particular, we explicitly determine some special constants including the Babu\v ska-Aziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element. The obtained results are expected to be widely used for a priori and a posteriori error estimations in adaptive computation and numerical verification of numerical solutions based on the triangular finite elements. We also give some numerical results for the error constants and for a posteriori estimates of some eigenvalues related to the error constants.

Keywords: FEM, error estimates, triangular finite elements, interpolation error constants.

Mathematics Subject Classification (2000): Primary 65N15, Secondary 65N30.