Approximation by sums of piecewise linear polynomials

We present two partitioning algorithms that allow a sum of piecewise linear polynomials over a number of overlaying convex partitions of the unit cube @W in R^d to approximate a function f@?W"p^3(@W) with the order N^-^6^/^(^2^d^+^1^) in the L"p-norm, where N is the total number of cells of all partitions, which makes a marked improvement over the N^-^2^/^d order achievable on a single convex partition. The gradient of f is approximated with the order N^-^3^/^(^2^d^+^1^). The first algorithm creates d convex partitions and relies on the knowledge of the eigenvectors of the average Hessians of f over the cells of an auxiliary uniform partition, whereas the second algorithm with d+12 convex partitions is independent of f. In addition, we also give an f-independent partitioning algorithm for a sum of d piecewise constants that achieves the approximation order N^-^2^/^(^d^+^1^).