SOLVERS FOR SELF-ADJOINT ELLIPTIC PROBLEMS IN IRREGULAR 2-DIMENSIONALDOMAINS

Software is provided for the rapid solution of certain types of ellipt ic equations in rectangular and irregular domains. Specifically, solut ions are found in two dimensions for the nonseparable self-adjoint ell iptic problem del.(gdelpsi) = f, where g and f are given functions of x and y, in two-dimensional polygonal domains with Dirichlet boundary conditions. Helmholtz and Poisson problems in polygonal domains and th e general variable coefficient problem (i.e., g not-equal 1) in a rect angular domain may be treated as special cases. The method of solution combines the use of the capacitance matrix method, to treat the irreg ular boundary, with an efficient iterative method (using the Laplacian as preconditioner) to deal with nonseparability. Each iterative step thus involves solving the Poisson equation in a rectangular domain. Th e package includes separate, easy-to-use routines for the Helmholtz pr oblem and the general problem in rectangular and general polygonal dom ains, and example driver routines for each. Both single- and double-pr ecision routines are provided. Second-order-accurate finite differenci ng is employed. Storage requirements increase approximately as p2 + n2 , where p is the number of irregular boundary points and where n is th e linear domain dimension. The preprocessing time (the capacitance mat rix calculation) varies as pn2log n, and the solution time varies as n 2log n. If the equations are to be solved repeatedly in the same geome try, but with different source or diffusion functions, the capacitance matrix need only be calculated once, and hence the algorithm is parti cularly efficient for such cases.