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/