One dimensional combination technique and its implementation

Abstract

This article introduces the 1D combination technique and its implementation with parallel programming. I discuss two primary features of the 1D combination technique: (1)~its reduction of computational cost, especially when combined with parallel programming and where high accuracy is required; and (2)~a resultant sacrifice of accuracy. However, the loss of the accuracy can be bounded thus reducing its significance. References C. Zenger. Sparse grids. in Parallel Algorithms for Partial Differential Equations, Proceedings of the Sixth GSAMMSeminar, Kiel, January 19--21, 1990, W. Hackbusch, ed., Braunschweig, 1991, Vieweg--Verlag. M. Griebel, M. Schneider, C. Zenger. A combination technique for the solution of sparse grid problems. PdeGroen, R.Buwens(Ed.). Iterative Methods in Linear Algebra, IMACS, Elsevier, North Holland, 1992, pp.263--281. http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.33.3530 C. Pflaum and A. Zhou. Error analysis of the combination technique. Numerische Mathematik, 84:327--350, 1999. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.54.1356 C. Pflaum. Convergence of the combination technique for second-order elliptic differential equations. SIAM J. Numer. Anal., 34(6):2431--2455, 1997. http://www.jstor.org/pss/2951959 U. Rude and A. Zhou. Multi-parameter extrapolation methods for boundary integral equaitons. Advances in Computational Mathematics, 9:173--190, 1998. http://www.springerlink.com/content/m31l311t32345607/