Box-spline–based approach to the formulation of numerical methods for partial differential equations

A multivariate box spline framework for the formulation of numerical methods for partial differential equations has been constructed. In particular, a fourth-order Galerkin method and a second-order collocation method were derived and applied to a test problem (classical Poisson equation on a square). The examples indicate that accuracy compares favorably with standard methods and the success of iterative schemes suggests an underlying stabilizing effect.     

A fully integrated high-throughput screening methodology for the discovery of new polyolefin catalysts: discovery of a new class of high temperature single-site group (IV) copolymerization catalysts

For the first time, new catalysts for olefin polymerization have been discovered through the application of fully integrated high-throughput primary and secondary screening techniques supported by rapid polymer characterization methods. Microscale 1-octene primary screening polymerization experiments combining arrays of ligands with reactive metal complexes M(CH(2)Ph)(4) (M = Zr, Hf) and multiple activation conditions represent a new high-throughput technique for discovering novel group (IV) polymerization catalysts. The primary screening methods described here have been validated using a commercially relevant polyolefin catalyst, and implemented rapidly to discover the new amide-ether based hafnium catalyst [eta(2)-(N,O)[bond](2-MeO[bond]C(6)H(4))(2,4,6-Me(3)C(6)H(2))N]Hf(CH(2)Ph)(3) (1), which is capable of polymerizing 1-octene to high conversion. The molecular structure of 1 has been determined by X-ray diffraction. Larger scale secondary screening experiments performed on a focused 96-member amine-ether library demonstrated the versatile high temperature ethylene-1-octene copolymerization capabilities of this catalyst class, and led to significant performance improvements over the initial primary screening discovery. Conventional one gallon batch reactor copolymerizations performed using selected amide-ether hafnium compounds confirmed the performance features of this new catalyst class, serving to fully validate the experimental results from the high-throughput approaches described herein.