A population of linear, second order, elliptic partial differential equations on rectangular domains. I, II

Authors:
John R. Rice, Elias N. Houstis and Wayne R. Dyksen

Journal:
Math. Comp. **36** (1981), 475-484

MSC:
Primary 65N99; Secondary 65M99

MathSciNet review:
606507

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Abstract: We present a population of 56 linear, two-dimensional elliptic partial differential equations (PDEs) suitable for evaluating numerical methods and software. Forty-two of the PDEs are parametrized which allows much larger populations to be made; 189 specific cases are presented here along with solutions (some are only approximate). Many of the PDEs are artificially created so as to exhibit various mathematical behaviors of interest; the others are taken from "real world" problems in various ways. The population has been structured by introducing measures of complexity of the operator, boundary conditions, solution and problem. The PDEs are first presented in mathematical terms along with contour plots of the 189 specific solutions. Machine-readable descriptions are given in Part 2; many of the PDEs involve lengthy expressions and about a dozen involve extensive tabulations of approximate solutions.

**[1]**R. E. Boisvert, E. N. Houstis & J. R. Rice, "A system for performance evaluation of partial differential equations software,"*IEEE Trans. Software Engrg.*, v. 5, 1979, pp. 418-425.**[2]**H. Crowder, R. S. Dembo & J. M. Mulvey, "On reporting computational experiments with mathematical software,"*ACM Trans. Math. Software*, v. 5, 1979, pp. 191-203.**[3]**S. C. Eisenstat and M. H. Schultz,*The complexity of partial differential equations*, Complexity of sequential and parallel numerical algorithms (Proc. Sympos., Carnegie-Mellon Univ., Pittsburgh, Pa., 1973) Academic Press, New York, 1973, pp. 271–282. MR**0398125****[4]**E. N. Houstis, R. E. Lynch, T. S. Papatheodorou & J. R. Rice, "Development, evaluation and selection of methods for elliptic partial differential equations,"*Ann. Assoc. Calcul. Analog.*, v. 11, 1975, pp. 98-105.**[5]**E. N. Houstis, R. E. Lynch, and J. R. Rice,*Evaluation of numerical methods for elliptic partial differential equations*, J. Comput. Phys.**27**(1978), no. 3, 323–350. MR**496854**, 10.1016/0021-9991(78)90014-1**[6]**E. N. Houstis & T. S. Papatheodorou, "Comparison of fast direct methods for elliptic problems,"*Advances in Computer Methods for Partial Differential Equations*II (R. Vishnevetsky, Ed.) IMACS, Rutgers University, New Brunswick, N.J., 1977, pp. 46-52.**[7]**E. N. Housis & T. S. Papatheodorou, "High order fast elliptic solver,"*ACM Trans. Math. Software*, v. 5, 1979, pp. 431-441.**[8]**E. N. Houstis & J. R. Rice, "An experimental design for the computational evaluation of elliptic partial differential equation solvers,"*The Production and Assessment of Numerical Software*(M. A. Hennell, Ed.), Academic Press, New York, 1980.**[9]**Robert E. Lynch and John R. Rice,*The Hodie method and its performance for solving elliptic partial differential equations*, Recent advances in numerical analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1978) Publ. Math. Res. Center Univ. Wisconsin, vol. 41, Academic Press, New York-London, 1978, pp. 143–175. MR**519061****[10]**J. R. Rice, "Methodology for the algorithm selection problem,"*Performance Evaluation of Numerical Software*(L. D. Fosdick, Ed.), North-Holland, Amsterdam, 1979, pp. 301-307.

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1981-0606507-2

Keywords:
Elliptic partial differential equations,
numerical methods,
software evaluation,
population of problems,
linear,
second order

Article copyright:
© Copyright 1981
American Mathematical Society