Algorithm 732: solvers for self-adjoint elliptic problems in irregular two-dimensional domains

Abstract

Software is provided for the rapid solution of certain types of elliptic equations in rectangular and irregular domains. Specifically, solutions are found in two dimensions for the nonseparable self-adjoint elliptic problem ∇·( g∇Ψ )= 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 the general variable coefficient problem (i.e., g≠1 ) in a rectangular domain may be treated as special cases. The method of solution combines the use of the capacitance matrix method, to treat the irregular boundary, with an efficient iterative method (using the Laplacian as preconditioner) to deal 2 with nonseparability. Each iterative step thus involves solving the Poisson equation in a rectangular domain. The package includes separate, easy-to-use routines for the Helmholtz problem and the general problem in rectangular and general polygonal domains, and example driver routines for each. Both single- and double-precision routines are provided. Second-order-accurate finite differencing is employed. Storage requirements increase approximately as p 2 + n 2 , where p is the number of irregular boundary points and where n is the linear domain dimension. The preprocessing time (the capacitance matrix calculation) varies as pn 2 log n , and the solution time varies as n 2 log n . If the equations are to be solved repeatedly in the same geometry, but with different source or diffusion functions, the capacitance matrix need only be calculated once, and hence the algorithm is particularly efficient for such cases.