Approximation by piecewise constants on convex partitions

Abstract

We show that the saturation order of piecewise constant approximation in L p norm on convex partitions with N cells is N − 2 / ( d + 1 ) , where d is the number of variables. This order is achieved for any f ∈ W p 2 ( Ω ) on a partition obtained by a simple algorithm involving an anisotropic subdivision of a uniform partition. This improves considerably the approximation order N − 1 / d achievable on isotropic partitions. In addition we show that the saturation order of piecewise linear approximation on convex partitions is N − 2 / d , the same as on isotropic partitions.