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 Omega in R-d to approximate a function f is an element of W-p(3) (Omega) with the order N-6/(2d+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/(2d+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 (2(d+1)) 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). (C) 2014 Elsevier Inc. All rights reserved.