Collocation software for second-order elliptic partial differential equations

ABSTRACT

We consider the collocation method for linear, second-order elliptic problems on rectangular and general two-dimensional domains. An overview of the method is given for general domains, followed by a discussion of the improved efficiencies and simplifications possible for rectangular domains. A very-high-level description is given of three specific collocation algorithms that use Hermite bicubic basic functions, (1) GENCOL (collocation on general two-dimensional domains), (2) HERMCOL (collocation on rectangular domains with general linear boundary conditions), and (3) INTCOL (collocation on rectangular domains with uncoupled boundary conditions). The linear system resulting from INTCOL has half the bandwidth of that from HERMCOL, which provides substantial benefit in solving the system. We provide some examples showing the range of applicability of the algorithms and some performance profiles illustrating their efficiency. Fortran implementations of these algorithms are given in the companion papers [10, 11].

REFERENCES

• 1: BALART, R., HOUSTIS, E. N., AND PAPATHEODOROU, T.S. On the iterative solution of collocation method equations. In 10th IMACS World Congress, (Montreal, Canada) vol. 1, R. Vishnevetsky, ed. 1982, pp. 98-100.
• 2: COURANT, R. Dirichlet's Principle, Conformal Mappings, and Minimal Surfaces. Vol. 3. Interscience, New York, 1950.
• 3: DYKSEN, W. R., HOUSTIS, E. N., LYNCH, R. E., AND RICE, J. R. The performance of the collocation and Galerkin methods with Hermite bicubics. SIAM J. Nurser. Anal. 21 (1984) 695-715.
• 4: DYKSEN, W. R., AND RICE, J.R. A New ordering scheme for the Hermite bicubic collocation equations. In Elliptic Problem Solvers If, G. Birkhoff and A. Schoenstat, Eds. Academic Press, New York, 1984, pp. 467-480.
• 5: Wayne R Dyksen , John R Rice, The importance of scaling for the Hermite bicubic collocation equations, SIAM Journal on Scientific and Statistical Computing, v.7 n.3, p.707-719, July 1986 [doi>10.1137/0907048]
• 6: GORbON, W. J., AND HALL, C.A. Construction of curvilinear coordinate systems and applications to mesh generation. Int. J. Numer. Meth. Eng. 7 (1973), 461-477.
• 7: HOUSTIS, E. N., LYNCH, R. E., PAPATHEODOROU, T. S., AND RICE, J.R. Evaluation of numerical methods for elliptic partial differential equations. J. Comput. Phy. 27 (1978), 323-350.
• 8: HOUSTIS, E. N., MITCHELL, W. F., AND PAPATHEODOROU, T.S. A Cl-collocation method for mildly nonlinear elliptic equations on general domains, in Advances in Computer Methods/or Partial Di//erential Equations II (R. Vishnevetsky, ed.) IMACS, Rutgers University, 1979, 13-17.
• 9: HOUSTIS, E. N., MITCHELL, W. F., AND PAPATHEODOROU, T. S. Performance evaluation of algorithms for mildly nonlinear elliptic problems. Int. J. Numer. Meth. Eng. 19 (1983), 665-709.
• 10: E. N. Houstis , W. F. Mitchell , J. R. Rice, GENCOL: collocation of general domains with bicubic hermite polynomials (Algorithm 637)., ACM Transactions on Mathematical Software (TOMS), v.11 n.4, p.413-415, Dec. 1985 [doi>10.1145/6187.6194]
• 11: E. N. Houstis , W. F. Mitchell , J. R. Rice, INTCOL and HERMCOL: collocation on rectangular domains with bicubic hermite polynomials (Algorithm 638)., ACM Transactions on Mathematical Software (TOMS), v.11 n.4, p.416-418, Dec. 1985 [doi>10.1145/6187.6195]
• 12: HOUSTIS, E. N., AND RICE, J.R. An experimental design for the computational evaluation of partial differential equation solvers. In Production and Assessment of Numerical Software, M. Delves and M. Hennell, Eds. Academic Press, London, pp. 57-66, 1980.
• 13: MASLIYAH, J. H., AND KUMAR, D. Application of orthogonal collocation on finite elements to a flow problem. Math. Comput. Simulation 22, (1980), 49-54.
• 14: PORITSKY, H. Calculation of flux distributions with saturation. AIEE Trans. 70 (1951}, 309-319.
• 15: John Rischard Rice, Numerical Methods, Software and Analysis, McGraw-Hill, Inc., New York, NY, 1983
• 16: RICE, J.R. Performance analysis of 13 methods to solve the Galerkin method equations. Linear Alg. Appl. 53, (1983), 533-546.
• 17: John R. Rice, Numerical Computation with General Two-Dimensional Domains, ACM Transactions on Mathematical Software (TOMS), v.10 n.4, p.443-452, Dec. 1984 [doi>10.1145/2701.356109]
• 18: John R. Rice, Algorithm 625: A Two-Dimensional Domain Processor, ACM Transactions on Mathematical Software (TOMS), v.10 n.4, p.453-462, Dec. 1984 [doi>10.1145/2701.356110]
• 19: John R. Rice , Ronald F. Boisvert, Solving elliptic problems using ELLPACK, Springer-Verlag New York, Inc., New York, NY, 1985
• 20: RICE, J. R., HOUSTIS, E. N., AND DYKSEN, W.R. A population of linear, second order elliptic partial differential equations on rectangular domains. Math. Comput. 36 (1981), 475-484.
• 21: ZIENKIEWlCZ, O. The Finite Element Method in Engineering Science. McGraw-Hill, London, 1971.

INDEX TERMS

Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.8 Partial Differential Equations
          Subjects: Finite element methods; Elliptic equations

General Terms:
Algorithms, Design, Measurement, Performance