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The question of adaptive mesh generation for approximation by splines hasbeen studied for a number of years by various authors. The results havenumerous applications in computational and discrete geometry, computer aidedgeometric design, finite element methods for numerical solutions of partialdifferential equations, image processing, and mesh generation for computergraphics, among others. In this paper we will investigate the questionsregarding adaptive approximation of C2 functions with arbitrary but fixedthroughout the domain signature by multilinear splines. In particular, we willstudy the asymptotic behavior of the optimal error of the weighted uniformapproximation by interpolating and quasi-interpolating multilinear splines.

Exact asymptotics of the uniform error of interpolation by multilinear splines