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For the Lagrange interpolation over a triangular domain, we propose anefficient algorithm to rigorously evaluate the interpolation error constantunder the maximum norm by using the finite element method (FEM). In solving theoptimization problem corresponding to the interpolation error constant, themaximum norm in the constraint condition is the most difficult part to process.To handle this difficulty, a novel method is proposed by combining theorthogonality of the interpolation associated to the Fujino--Morley FEM spaceand the convex-hull property of the Bernstein representation of functions inthe FEM space. Numerical results for the lower and upper bounds of theinterpolation error constant for triangles of various types are presented toverify the efficiency of the proposed method.

Error constant estimation under the maximum norm for linear Lagrange interpolation