A Population of Linear, Second Order, Elliptic Partial Differential Equations on Rectangular Domains. Part 1

摘要

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 parameterized 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, MRC Technical Summary Report no. 2079; many of the PDEs involve lengthy expressions and about a dozen involve extensive tabulations of approximate solutions. (Author)