A Schwarz splitting variant of cubic spline collocation methods for elliptic PDEs

ABSTRACT

We consider the formulation of the Schwarz alternating method for a new class of elliptic cubic spline collocation discretization schemes. The convergence of the method is studied using Jacobi and Gauss-Seidel iterative methods for implementing the interaction among subdomains. The Schwarz Cubic Spline Collocation (SCSC) method is formulated for hypercube architectures and implemented on the NCUBE (128 processors) machine. The performance and convergence of the hypercube SCSC algorithm is studied with respect to domain partition and subdomain overlapping area. The numerical results indicate that the partition and mapping of the SCSC on the NCUBE is almost optimal while the speedup obtained is similar to other domain decomposition techniques.

REFERENCES

• Bjor 86: Petter E Bjorstad , Olof B Widlund, Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM Journal on Numerical Analysis, v.23 n.6, p.1097-1120, Dec. 1986 [doi>10.1137/0723075]
• Ehrl 86: Louis W Ehrlich, The Numerical Schwarz alternating procedure and SOR, SIAM Journal on Scientific and Statistical Computing, v.7 n.3, p.989-993, July 1986 [doi>10.1137/0907066]
• Hous 87: E. N. Housfis, J. R. Rice and E. A. Vavalis, Parallelization of a new class of cubic $pline collocation methods, in: Advances in Computer Methods for Partial Differential Equations, Vol. VI, (R. Vichnevetsky, R. S. Stepleman, editor), 167-174.
• Hous 88a: E. N. Houstis , E. A. Vavalis , J. R. Rice, Convergence of O(h4) cubic spline collocation methods for elliptic partial differential equations, SIAM Journal on Numerical Analysis, v.25 n.1, p.54-74, February 1988 [doi>10.1137/0725006]
• Hous 88b: E. N. Houstis, J. R. Rice, E. A. Vavalis and A. Hadjidimos, Convergence of Schwarz Decomposition for a Class of Cubic $pline Collocation Methods. (In preparation).
• Keye 87: David E. Keyes , William D. Gropp, A comparison of domain decomposition techniques for elliptic partial differential equations and their parallel implementation, SIAM Journal on Scientific and Statistical Computing, v.8 n.2, p.166-202, March 1, 1987 [doi>10.1137/0908020]
• Luba 84: B. Lubachevsky and D. Mitra, Chaotic parallel computations of ftxed points of nonnegative matrices of unit spectral radius, iEEE, 109-115.
• Meie 86: U Meier, Two parallel SOR variants of the Schwarz alternating procedure, Parallel Computing, v.3 n.3, p.205-215, July 1986 [doi>10.1016/0167-8191(86)90020-7]
• Reed 84: D. A. Reecl and M. L. Patrick, A model of asynchronous iterative algorithms for solving large, sparse, linear sy$. terns, IEF_..E, 402--409.
• Rice 85: John R. Rice , Ronald F. Boisvert, Solving elliptic problems using ELLPACK, Springer-Verlag New York, Inc., New York, NY, 1985
• Rodr 84: G. Rodrigue and J. Simon, Jacobi splittings and the method of overlapping domains for solving elliptic PDEs, in: Advances in Computer Methods for Partial Differential Equations, Vol. V, (R. Vichnevetsky, R. S. Stepleman, editor), 383-386.
• Rodr 86: G. Rodrigue, Some ideas for decompoMng the domain of elliptic partial differential equations in the Schwarz process. Comm. Appl. Num. Meth., Vol, 2, 245-249.
• Tang 87: W.P. Tang, Schwarz splitting, a model of parallel computations, Ph.D. Thesis, Stanford University.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.8 Partial Differential Equations
          Subjects: Elliptic equations

General Terms:
Algorithms, Theory