SOLVING LAPLACE'S EQUATION IN 3D ON A SMOOTH SURFACE This subdirectory contains programs for solving the interior Dirichlet problem for Laplace's equation in 3D. The programs are based on using a Galerkin method with spherical polynomials as approximants. The method is described in the paper K. Atkinson (1985) Algorithm 629: An Integral Equation Program for Laplace's Equation in Three Dimensions, ACM Trans. Math. Soft. 11, pp. 85-96. This program is suitable only for regions which are simply connected and which have a smooth boundary that is homeomorphic to the unit sphere. Examples are given in the program. The problem solving process is divided into two parts. First the Galerkin coefficients are computed, using the program "gnrt.f"; and second, these are input into the program "laplace_3d.f" which does the actual solution. The major cost of the method is in the generation of the Galerkin coefficients, as is explained in the article cited above. CALCULATING THE GALERKIN COEFFICIENTS: The files gnrt.f print_coeff.f basis_num.f input_file_name gnrt.data are used in connection with generating the Galerkin coefficients for solving the interior Dirichlet problem for Laplace's equation. gnrt.f : This generates the Galerkin coefficients. It reads from the standard input unit the name of the file containing the data for "gnrt.f". If "gnrt.exe" is the run-file for "gnrt.f", then one could use gnrt.exe < input_file_name > file.out & to run the program in background. "gnrt.f" produces two output files of the Galerkin coefficients, one in decimal and the other in formatfree form. The latter is used as input to "laplace_3d.f". The first is a look at the Galerkin coefficients in decimal form for the maximum degree specified by the use in running "gnrt.f". print_coeff.f: This produces a more readable copy of the Galerkin coefficients associated with a particular degree. It uses as input the formatfree file produced by "gnrt.f". basis_num.f: This produces a correspondence between the linear indexing of the coefficients and the usual parameters (n,m,l) associated with using spherical harmonics. gnrt.data: This is a sample data file of input parameters to "gnrt.f". To see the meaning of the numbers in this file, look at the file "gnrt.data-model". CALCULATING THE SOLUTION OF THE DIRICHLET PROBLEM: The files dirichlet_3d.f dirichlet.data are used in solving the Dirichlet problem for Laplace's equation. The method of solution is to represent the unknown potential as a double layer potential, and to then solve for the double layer density function. After this is found, the solution is then evaluated at given interior data points, by numerical integration. The file "dirichlet.data" contains data for running "dirichlet_3d.f". For the meaning of this data file, see "dirichlet.data-model". OTHER BOUNDARY VALUE PROBLEMS: We have also extended these programs to other boundary value problems, as is discussed in articles referenced in the article cited above. This includes the exterior Dirichlet problem and the Neumann problem, both interior and exterior. These are not in a form as finished as those given here; but they are available from the author.